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A LINNTHIT- PRIVATE SCHOOL A LINNTHIT- PRIVATE SCHOOL ……………………………………………………………………………………………………………………………………………………………………………………………………………………………………………

Koaunglwinoo ( thufonesarr-education website ) Page | 1

B

H

M

EXERCISE 7.1

1.Let Maung Ba = B , Maung Hla = H , Maung Mya = M ,

Ma Ni = N , Ma Yi = Y , Ma Thi = T , Ma Si = S

President Vice President Possible Outcomes

N ( B, N )

Y ( B, Y )

T ( B, T )

S ( B, S )

N ( H, N )

Y ( H, Y )

T ( H, T )

S ( H, S )

N ( M, N )

Y ( M, Y )

T ( M, T )

S ( M, S )

P ( Maung Ba is to be selected for presidents ) = ?

The set of all possible outcomes = { ( B , N ) , ( B, Y ) , ( B, T ) ,( B, S ) ,( H, N ) ,( H,

Y ), ( H, T )

( H, S ) , ( M, N ) , ( M, Y ) , ( M, T ) , (

M, S ) }

Number of possible outcomes = 12

The set of favourable outcomes = { ( B , N ) , ( B, Y ) , ( B, T ) ,( B, S ) }

Number of favourable outcomes = 4

P ( A ) = 𝐍𝐮𝐦𝐛𝐞𝐫 𝐨𝐟 𝐟𝐚𝐯𝐨𝐮𝐫𝐚𝐛𝐥𝐞 𝐨𝐮𝐭𝐜𝐨𝐦𝐞𝐬

𝐍𝐮𝐦𝐛𝐞𝐫 𝐨𝐟 𝐩𝐨𝐬𝐬𝐢𝐛𝐥𝐞 𝐨𝐮𝐭𝐜𝐨𝐦𝐞𝐬

P ( Maung Ba is to be selected for presidents ) = 𝟒

𝟏𝟐 =

𝟏

𝟑

---------------------------------------------------------------------

2. replaced ပစာၦ

1st Choice 2nd Choice Possible Outcomes

2 ( 2 , 2 )

3 ( 2 , 3 )

2 4 ( 2 , 4 )

5 ( 2 , 5 )

9 ( 2 , 9 )

2 ( 3 , 2 )

3 ( 3 , 3 )

3 4 ( 3 , 4 )

5 ( 3 , 5 )

9 ( 3 , 9 )

2 ( 4 , 2 )

3 ( 4 , 3 )

4 4 ( 4 , 4 )

5 ( 4 , 5 )

9 ( 4 , 9 )

2 ( 5 , 2 )

3 ( 5 , 3 )

5 4 ( 5 , 4 )

5 ( 5 , 5 )

9 ( 5 , 9 )

2 ( 9 , 2 )

3 ( 9 , 3 )

9 4 ( 9 , 4 )

5 ( 9 , 5 )

9 ( 9 , 9 )

The set of all possible outcomes ={ ( 2 , 2 ),( 2 , 3 ),( 2 , 4 ) ,( 2 , 5 ), ( 2 , 9 )

,( 3 , 2 ),( 3 , 3 ) ,( 3 , 4 ),( 3 , 5 ),( 3 , 9 ) ,( 4 , 2 ) ,( 4 , 3 ) ,( 4 , 4 ) ,( 4 , 5 )

,( 4 , 9 ),( 5 , 2 ) ,( 5 , 3 ) ,( 5 , 4 ) ,( 5 , 5 ) ,( 5 , 9 ) , ( 9 , 2 ),( 9 , 3 ),( 9 , 4 )

,( 9 , 5 ),( 9 , 9 )}


( a ) P ( getting two prime numbers ) = ?



**( ေရ႕ကနးေရာ ေနာကကနးေရာ prime number မား ျဖစေနမည ျဖစတနစြမး )

The set of favourable outcomes ={ ( 2 , 2 ),( 2 , 3 ),( 2 , 5 ),( 3 , 2 ),( 3 , 3 )

,( 3 , 5 ) ,( 5 , 2 ) ,( 5 , 3 ),( 5 , 5 ) }




P ( getting two prime numbers ) = 𝟗

𝟐𝟓

( b ) P ( getting two odd numbers ) = ? ( ေရ႕ကနး ေနာကကနးမကနးမား ျဖစေနမည ျဖစတနစြမး )

The set of favourable outcomes ={ ( 3 , 3 ),( 3 , 5 ),( 3 , 9 ),( 5 , 3 ),( 5 , 5 )

,( 5 , 9 ) ,( 9 , 3 ),( 9 , 5 ),( 9 , 9 ) }


P ( getting two prime numbers ) = 𝟗

𝟐𝟓

( c ) P ( getting a pair of numbers where the sum is a prime numbers ) = ?

**( ေရ႕ကနးႏင ေနာကကနးေပါငးျခငးတနဖး prime number ျဖစေနမည ျဖစတနစြမး )

The set of favourable outcomes ={ ( 2 , 3 ),( 2 , 5 ),( 2 , 9 ) ,( 3 , 2 ),( 3 , 4 )

,( 4 , 3 ),( 4 , 9 ),( 5 , 2 ),( 9 , 2 ),( 9 , 4 ) }


P ( getting a pair of numbers where the sum is a prime numbers ) = 𝟏𝟎

𝟐𝟓

---------------------------------------------------------------------

3. not replaced ပစာၦ


3 ( 2 , 3 )

4 ( 2 , 4 )

5 ( 2 , 5 )

9 ( 2 , 9 )

2 ( 3 , 2 )

4 ( 3 , 4 )

5 ( 3 , 5 )

9 ( 3 , 9 )

2 ( 4 , 2 )

3 ( 4 , 3 )

5 ( 4 , 5 )

9 ( 4 , 9 )

2 ( 5 , 2 )

3 ( 5 , 3 )

4 ( 5 , 4 )

9 ( 5 , 9 )

2 ( 9 , 2 )

3 ( 9 , 3 )

4 ( 9 , 4 )

5 ( 9 , 5 )

The set of all possible outcomes ={ ( ( 2 , 3 ),( 2 , 4 ),( 2 , 5 ),( 2 , 9 )

,( 3 , 2 ),( 3 , 4 ),( 3 , 5 ),( 3 , 9 ) ,( 4 , 2 ) ,( 4 , 3 ),( 4 , 5 ),( 4 , 9 )

,( 5 , 2 ) ,( 5 , 3 ) ,( 5 , 4 ),( 5 , 9 ), ( 9 , 2 ),( 9 , 3 ),( 9 , 4 ),( 9 , 5 )}


( a ) P ( getting two prime numbers ) = ?

The set of favourable outcomes ={ ( 2 , 3 ),( 2 , 5 ),( 3 , 2 )

, ( 3 , 5 ) ( 5 , 2 ) ,( 5 , 3 ) }




P ( getting two prime numbers ) = 𝟔

𝟐𝟓

( b ) P ( getting two odd numbers ) = ?

The set of favourable outcomes ={ ( 3 , 5 ),( 3 , 9 ),( 5 , 3 )

,( 5 , 9 ) ,( 9 , 3 ),( 9 , 5 ) }


P ( getting two prime numbers ) = 𝟔

𝟐𝟓

( c ) P ( getting a pair of numbers where the sum is a prime numbers ) = ?

The set of favourable outcomes ={ ( 2 , 3 ),( 2 , 5 ),( 2 , 9 ) ,( 3 , 2 ),( 3 , 4 )

2

3

4

5

9



,( 4 , 3 ),( 4 , 9 ),( 5 , 2 ),( 9 , 2 ),( 9 , 4 ) }


P ( getting a pair of numbers where the sum is a prime numbers ) = 𝟏𝟎

𝟐𝟓

---------------------------------------------------------------------

4. not replaced ပစာၦ

Let the 2 blue marble = b1 , b

2 , 1 red marble = r , 1 yellwo marble = y


b2 ( b

1 , b

2 )

b1 r ( b

1 , r )

y ( b1 , y )

b1 ( b

2 , b

1 )

b2 r ( b

2 , r )

y ( b2 , y )

b1 ( r , b

1 )

r b2 ( r , b

2 )

y ( r , y )

b1 ( r , b

1 )

y b2 ( r , b

2 )

r ( y , r )

The set of all possible outcomes = {( b1 , b

2 ),( b

1 , r ),( b

1 , y ),( b

2 , b

1 )

,( b2 , r ), ( b

2 , y ),( r , b

1 ),( r , b

2 ),( r , y ),( r , b

1 ),( r , b

2 ),( y , r ) }


( a ) P ( choosing 2 blue marbles ) = ?

The set of favourable outcomes = { ( b1 , b

2 ),( b

2 , b

1 ) }


P ( A ) = 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑓𝑎𝑣𝑜𝑢𝑟𝑎𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠

𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠

P ( choosing 2 blue marbles ) = 𝟐

𝟏𝟐 =

𝟏

𝟔

( b )P ( choosing 2 different colours ) = ?

The set of favourable outcomes = { ( b1 , r ),( b

1 , y ),( b

2 , r ), ( b

2 , y )

,( r , b1 ),( r , b

2 ) ,( r , y ),( r , b

1 ),( r , b

2 ),( y , r ) }


P ( choosing 2 different colours ) = 𝟏𝟎

𝟏𝟐 =

𝟓

𝟔

---------------------------------------------------------------------

5. 1st Spin 2nd Spin Possible Outcomes

B ( B , B )

B R ( B , R )

Y ( B , Y )

B ( R , B )

R R ( R , R )

Y ( R , Y )

B ( Y , B )

Y R ( Y , R )

Y ( Y , Y )

ျမားလညသည ပစာၦတငး

သည replaced ပစာၦ

မားႏင သေဘာအတ

တပငျဖစသည။



The set of all possible outcomes = { ( B , B ),( B , R ),( B , Y ),( R , B ),( R , R )

,( R , Y ),( Y , Y ),( Y , R ),( Y , B ) }


( a ) P ( not spinning red first ) = ? ( ေရ႕အေရာင အနမျဖစသည ျဖစတနစြမး )

The set of favourable outcomes ={( B , B ),( B , R ),( B , Y ),( Y , Y ),( Y , R ),( Y , B ) }




P ( not spinning red first ) = 𝟔

𝟗 =

𝟐

𝟑

( a ) P ( spinning 2 different colours ) = ?

The set of favourable outcomes ={( B , R ),( B , Y ),( R , B ),( R , Y ),( Y , B ),( Y , R ) }


P (spinning 2 different colours ) = 𝟔

𝟗 =

𝟐

𝟑

---------------------------------------------------------------------

6. ဆေပးျခငး ဥကဌ ေရြးျခငး ျဖစရပမားသည not replaced ျဖစရပမား ျဖစၾကပါသညၤ။

Let Maung Maung = Maung , Maung Mya = Mya , Ma Hla = Hla , Ma Khin = Khin

1st Prize 2nd Prize Possible Outcomes

Mya ( Maung , Mya )

Maung Hla ( Maung , Hla )

Khin ( Maung , Khin )

Maung ( Mya , Maung )

Mya Hla ( Mya , Hla )

Khin ( Mya , Khin )

Maung ( Hla , Maung )

Hla Mya ( Hla , Mya )

Khin ( Hla , Khin )

Maung ( Khin , Maung )

Khin Mya ( Khin , Mya )

Hla ( Khin , Hla )

The set of all possible outcomes ={ ( Maung , Mya ),( Maung , Hla ), ( Maung , Khin )

,( Mya , Maung ) ,( Mya , Hla ),( Mya , Khin )

,( Hla , Maung ),( Hla , Mya ), ( Hla , Khin )

,( Khin , Maung ), ( Khin , Mya ), ( Khin , Hla ) }


P ( Maung Mya and Ma Khin both win prizes ) = ?



The set of favourable outcomes = { ( Mya , Khin ) , ( Khin , Mya ) }


P ( Maung Mya and Ma Khin both win prizes ) = 2

12 =

1

6

---------------------------------------------------------------------

7. ဒဂၤြးျပားေျမာကျခငးျဖစရပမားသည replaced ျဖစရပမား ျဖစၾကပါသညၤ။

1st Toss 2nd Toss 2nd Toss Possible Outcomes

H ( H , H , H )

T ( H , H , T )

H ( H , T , H )

T ( H , T , T )

H ( T , H , H )

T ( T , H , T )

H ( T , T , H )

T ( T , T , T )

H

T

H

T

H

T



The set of all possible outcomes = { ( H , H , H ),( H , H , T ),( H , T , H )

,( H , T , T ),( T , H , H ),( T , H , T ),( T , T , H ), ( T , T , T ) }


( a ) P ( getting exactly one head ) = ?

( သးၾကမးေျမာကတာမာ ေခါငးတစၾကမတညး တတကကပါတ ျဖစတနစြမး )

The set of favourable outcomes = { ( H , T , T ), ( T , H , T ), ( T , T , H ) }


P ( A ) = 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑓𝑎𝑣𝑜𝑢𝑟𝑎𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚 𝑒𝑠


P ( getting exactly one head ) = 3

8

( b ) P ( getting no head ) = ?

( သးၾကမးေျမာကတာမာ ေခါငးတစၾကမမ မကတ ျဖစတနစြမး )

The set of favourable outcomes = { ( T , T , T ) }


P ( getting no head ) = 1

8

---------------------------------------------------------------------

8.

Let boy = B , girl = G

1st Child 2nd Child 2nd Child Possible Outcomes

B ( B , B , B )

G ( B , B , G )

B ( B , G , B )

G ( B , G , G )

B ( G , B , B )

G ( G , B , G )

B ( G, G , B )

G ( G , G , G )

The set of all possible outcomes = { ( B , B , B ),( B , B , G ),( B , G , B )

,( B , G , G ),( G , B , B ),( G , B , G ),( G , G , B ),( G , G , G ) }


( a ) P ( the first two children are boys ) = ?

The set of favourable outcomes = { ( B , B , B ),( B , B , G ) }



𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑜𝑠𝑠𝑖𝑏 𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠

P (the first two children are boys ) = 2

8 =

1

4

( b ) P (the first two children are boys ) = ?

( သးၾကမးေျမာကတာမာ ေခါငးတစၾကမမ မကတ ျဖစတနစြမး )

The set of favourable outcomes = { ( B , B , B ),( G , B , B ) }


P ( getting no head ) = 2

8 =

1

4

---------------------------------------------------------------------

B

G

B

G

B

G

သးေယာကေမြးတာမာ

ပထမႏစေယာက

ေယာကာေလး ျဖစတနစြမး



8.

Tossing Co[n 2nd

Thrown 2nd

Thrown Possible Outcomes Possible Outcomes 1 ( H , 1 , 1 ) ( T , 1 , 1 ) 2 ( H , 1 , 2 ) ( T , 1 , 2 ) 3 ( H , 1 , 3 ) ( T , 1 , 3 ) 4 ( H , 1 , 4 ) ( T , 1 , 4 ) 5 ( H , 1 , 5 ) ( T , 1 , 5 ) 6 ( H , 1 , 6 ) ( T , 1 , 6 ) 1 ( H , 2 , 1 ) ( T , 2 , 1 ) 2 ( H , 2 , 2 ) ( T , 2 , 2 ) 3 ( H , 2 , 3 ) ( T , 2 , 3 ) 4 ( H , 2 , 4 ) ( T , 2 , 4 ) 5 ( H , 2 , 5 ) ( T , 2 , 5 ) 6 ( H , 2 , 6 ) ( T , 2 , 6 ) 1 ( H , 3 , 1 ) ( T , 3 , 1 ) 2 ( H , 3 , 2 ) ( T , 3 , 2 ) 3 ( H , 3 , 3 ) ( T , 3 , 3 ) 4 ( H , 3 , 4 ) ( T , 3 , 4 ) 5 ( H , 3 , 5 ) ( T , 3 , 5 ) 6 ( H , 3 , 6 ) ( T , 3 , 6 ) 1 ( H , 4 , 1 ) ( T , 4 , 1 ) 2 ( H , 4 , 2 ) ( T , 4 , 2 ) 3 ( H , 4 , 3 ) ( T , 4 , 3 ) 4 ( H , 4 , 4 ) ( T , 4 , 4 ) 5 ( H , 4 , 5 ) ( T , 4 , 5 ) 6 ( H , 4 , 6 ) ( T , 4 , 6 ) 1 ( H , 5 , 1 ) ( T , 5 , 1 ) 2 ( H , 5 , 2 ) ( T , 5 , 2 ) 3 ( H , 5 , 3 ) ( T , 5 , 3 ) 4 ( H , 5 , 4 ) ( T , 5 , 4 ) 5 ( H , 5 , 5 ) ( T , 5 , 5 ) 6 ( H , 5 , 6 ) ( T , 5 , 6 ) 1 ( H , 6 , 1 ) ( T , 6 , 1 ) 2 ( H , 6 , 2 ) ( T , 6 , 2 ) 3 ( H , 6 , 3 ) ( T , 6 , 3 ) 4 ( H , 6 , 4 ) ( T , 6 , 4 ) 5 ( H , 6 , 5 ) ( T , 6 , 5 ) 6 ( H , 6 , 6 ) ( T , 6 , 6 )

The set of all possible outcomes = { ( H , 1 , 1 ),( H , 1 , 2 ),( H , 1 , 3 ),( H , 1 , 4 )

,( H , 1 , 5 ),( H , 1 , 6 ),( H , 2 , 1 ),( H , 2 , 6 ),( H , 2 , 3 ),( H , 2 , 4 ),( H , 2 , 5 ),( H , 2 , 2 ),( H , 3 , 1 )

.( H , 3 , 2 ).( H , 3 , 3 ).( H , 3 , 4 ),( H , 3 , 5 ),( H , 3 , 6 ),( H , 4 , 1 ),( H , 4 , 2 ),( H , 4 , 3 ),( H , 4 , 4 )

,( H , 4 , 5 ),( H , 4 , 6 ),( H , 5 , 1 ),( H , 5 , 2 ),( H , 5 , 3 ),( H , 5 , 4 ),( H , 5 , 5 ),( H , 5 , 6 ),( H , 6 , 1 )

,( H , 6 , 2 ),( H , 6 , 3 ),( H , 6 , 6 ),( H , 6 , 5 ),( H , 6 , 4 ),( T , 1 , 1 ),( T , 1 , 2 ),( T , 1 , 3 ),( T , 1 , 4 )

,( T , 1 , 6 ),( T , 1 , 5 ),( T , 2 , 1 ),( T , 2 , 2 ),( T , 2 , 3 ),( T , 2 , 4 ),( T , 2 , 5 ),( T , 2 , 6 ),( T , 3 , 1 )

,( T , 3 , 2 ),( T , 3 , 3 ),( T , 3 , 4 ),( T , 3 , 5 ),( T , 3 , 6 ),( T , 4 , 1 ),( T , 4 , 2 ),( T , 4 , 3 ),( T , 4 , 4 )

,( T , 4 , 5 ),( T , 4 , 6 ),( T , 5 , 1 ),( T , 5 , 2 ),( T , 5 , 3 ),( T , 5 , 4 ),( T , 5 , 5 ),( T , 5 , 6 ),( T , 6 , 1 )

,( T , 6 , 2 ),( T , 6 , 6 ),( T , 6 , 4 ),( T , 6 , 5 ),( T , 6 , 3 ) }


P ( Head and 6 turn up ) = ?

The set of favourable outcomes = { ( H , 1 , 6 ),( H , 2 , 6 ),( H , 3 , 6 )

,( H , 4 , 6 ),( H , 5 , 6 ), ( H , 6 , 1 )

,( H , 6 , 2 ),( H , 6 , 3 ),( H , 6 , 4 )

,( H , 6 , 5 ), ( H , 6 , 6 ) }




P (Head and 6 turn up ) = 11

72

---------------------------------------------------------------------

Example ( 1 ) Black dies ( or ) 2nd die

The set of all possible outcome = { ( 1 , 1 ), ( 1 , 2 ), ( 1 , 3 ),( 1 , 4 ),( 1 , 5 )

,( 1 , 6 ),( 2 , 1 ),( 2 , 2 ),( 2 , 3 ),( 2 , 4 ),( 2 , 5 ),( 2 , 6 ),( 3 , 1 ),( 3 , 2 )

,( 3 , 3 ),( 3 , 4 ),( 3 , 5 ),( 3 , 6 ),( 4 , 1 ),( 4 , 2 ),( 4 , 3 ),( 4 , 4 ),( 4 , 5 )

1 2 3 4 5 6

1

2

3

4

5

6

( 1 , 1 ) ( 1 , 2 ) ( 1 , 3 ) ( 1 , 4 ) ( 1 , 5 ) ( 1 , 6 )

( 2 , 1 ) ( 2 , 2 ) ( 2 , 3 ) ( 2 , 4 ) ( 2 , 5 ) ( 2 , 6 )

( 3 , 1 ) ( 3 , 2 ) ( 3 , 3 ) ( 3 , 4 ) ( 3 , 5 ) ( 3 , 6 )

( 4 , 1 ) ( 4 , 2 ) ( 4 , 3 ) ( 4 , 4 ) ( 4 , 5 ) ( 4 , 6 )

( 5 , 1 ) ( 5 , 2 ) ( 5 , 3 ) ( 5 , 4 ) ( 5 , 5 ) ( 5 , 6 )

( 6 , 1 ) ( 6 , 2 ) ( 6 , 3 ) ( 6 , 4 ) ( 6 , 5 ) ( 6 , 6 )

1

1

1

1

1

1

H

or

T

P ( not A ) = 1 - P ( A )

P ( A or B ) = P ( A ) + P ( B ) ( mutually exclusive outcomes )

P ( A and B ) = P ( a ) x P ( B ) ( independent outcomes )

Blue die

Or

1st die



,( 4 , 6 ),( 5 , 1 ),( 5 , 2 ),( 5 , 3 ),( 5 , 4 ),( 5 , 5 ),( 5 , 6 ),( 6 , 1 ),( 6 , 2 )

,( 6 , 3 ),( 6 , 4 ),( 6 , 5 ),( 6 , 6 ) }

Number of possible outcome = 36

( 1 ) = P ( 5 ) = ? ( ႏစခေပါငး 5 ရမည ျဖစတနစြမး )

The set of favourable outcomes = { ( 1,4 ),( 2,3 ),( 3,2 ),( 4,1 ) }



𝑁𝑢𝑚𝑏 𝑒𝑟 𝑜𝑓 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠

P ( 5 ) = 4

36 =

1

9

---------------------------------------------------------------------

( 2 ) P ( not 5 ) = ?

P ( not 5 ) = 1 – P ( 5 ) = 1 - 1

9 =

8

9

---------------------------------------------------------------------

( 3 ) P ( 5 or 10 ) = ?

ႏစခေပါငး 5 ကသည ျဖစတနစြမး ( သ႔ ) 10 ကသည ျဖစတနစြမး

The set of all possible outcomes = { ( 1,4 ),( 2,3 ),( 3,2 ),( 4,1 ),( 4,6 ),( 5,5 ),( 6,4 ) }


P ( 5 or 10 ) = 7

36

Mutually exclusive formula အရ တြကမညဆလင

P ( 5 or 10 ) = P ( 5 ) + P ( 10 )

The set of all possible outcomes = { ( 4,6 ),( 5,5 ),( 6,4 ) }


P ( 10 ) = 3

36

P ( 5 or 10 ) = P ( 5 ) + P ( 10 ) = 4

36 +

3

36 =

7

36

---------------------------------------------------------------------

( 4 )P ( Blue 4 ) = ?

အျပာေရာငအစာတ 4 ကမည ျဖစတနစြမး = ( 4,1 ),( 4,2 ),( 4,3 ),( 4,4 ) ,( 4,5 ),( 4,6 )

The set of all possible outcomes = { ( 4,1 ),( 4,2 ),( 4,3 ),( 4,4 ) ,( 4,5 ),( 4,6 ) }


P ( Blue 4 ) = 6

36 =

1

6

---------------------------------------------------------------------

( 5 )P ( Black 5 ) = ?

အနကေရာငအစာတ 4 ကမည ျဖစတနစြမး = ( 1,5 ),( 2,5 ),( 3,5 ),( 4,5 ) ,( 5,5 ),( 6,5 )

The set of all possible outcomes = { ( 1,5 ),( 2,5 ),( 3,5 ),( 4,5 ) ,( 5,5 ),( 6,5 ) }


P ( Black 5 ) = 6

36 =

1

6

---------------------------------------------------------------------

( 6 ) P ( Blue 4 and Black 5 ) = ?

အျပာေရာငအစာတ 4 ႏင အနကေရာငအစာတ 4 တၿပငတညး ကမည ျဖစတနစြမး =

The set of all possible outcomes = { ( 4,5 ) }


P (Blue 4 and Black 5 ) = 1

36

Independent event formula အရ

P ( Blue 4 and Black 5 ) = P ( Blue 4 ) x P (Black 5 ) = 1

6 x

1

6 =

1

36

မတခက။ ။Total score ေမးလင အနညးဆး 2 မ အမားဆး 12 အထ ရနင။

အစာတးတစတးခငးေပၚရ တနဖးေမးလင အနညးဆး 1 မ အမားဆး 6 အထ ရနင။

ႏစခေပါငး 5 ကသည ျဖစတနစြမး = ( 1,4 ),( 2,3 ),( 3,2 ),( 4,1 )

ႏစခေပါငး 10 ကသည ျဖစတနစြမး = ( 4,6 ),( 5,5 ),( 6,4 )

ႏစခေပါငး 5 ကသည ျဖစတနစြမး ( သ႔ ) 10 ကသည ျဖစတနစြမး

= ( 1,4 ),( 2,3 ),( 3,2 ),( 4,1 ) ,( 4,6 ),( 5,5 ),( 6,4 )



Exercise 7.2

( 7 ) ( a ) P ( a 1 on the blue dice ) = ?

black dice ဘာျဖစျဖစ blue dice 1 ကမည ျဖစရပ

Number of possible outcomes ={ ( 1,2 ),( 1,3 ),( 1,4 ),( 1,5 ),( 1,6 ) }

The set of all possible outcomes = 6

P ( a 1 on the blue dice ) = 6

36 =

1

6

---------------------------------------------------------------------

( b ) P ( a 1 on the blue dice or a 6 on the blue dice ) = P ( blue 1 or blue 6 ) = ?

black dice ဘာျဖစျဖစ blue dice 1 ကမည ျဖစရပ (သ )

black dice ဘာျဖစျဖစ blue dice 6 ကမည ျဖစရပ

The set of favourable outcomes = { ( 1 ,1 ),( 1,2 ),( 1,3 ),( 1,4 ),( 1,5 ),( 1,6 ),( 6,1 )

,( 6,2 ),( 6,3 ),( 6,4 ),( 6,5 ),( 6,6 ) }


P ( blue 1 or blue 6 ) = 12

36 =

1

3

---------------------------------------------------------------------

( c ) P ( a 1 on the blue dice or a 5 on the black dice ) = P ( blue 1 or black 5 ) =?

black dice ဘာျဖစျဖစ blue dice 1 ကမည ျဖစရပ (သ )

blue dice ဘာျဖစျဖစ black dice 5 ကမည ျဖစရပ

The set of favourable outcomes = { ( 1 ,1 ),( 1,2 ),( 1,3 ),( 1,4 ),( 1,5 ),( 1,6 ),( 1,5 )

,( 2,5 ),( 3,5 ),( 4,5 ),( 5,5 ),( 6,5 ) }


P ( blue 1 or black 6 ) = 12

36 =

1

3

---------------------------------------------------------------------

( 9 ) Eg ( 1 ) - 4,5,6 ႏငအတတပင။ ပထမေမးခြနးအတြကတြရာေပး ၊ ဒတယေမးခြနးက

သးခငးတာျဖစ၍ ခြရာေပး။

( 10 ) ပထမေမးခြနးမာ Independent formula မသးပ ရာခငးျခငးျဖစသည။

P ( blue 1 and black number greater than 4 ) =

P ( blue 1 and black number > 4 ) = ?

blue die 1 ႏင Black die 4 ထကႀကးသည တနဖးကမည ျဖစရပ

The set of favourable outcomes = { ( 1,5 ),( 1,6 ) }


P ( blue 1 and black number > 4 ) = 2

36 =

1

18

ဒတယေမးခြနးမာ ထေမးခြနးကပင Independent formula သး၍ ရာခငးျခငးျဖစသည။

P ( blue 1 and black number > 4 ) = P ( blue 1 ) + P ( black number > 4 )

P ( blue 1 ) = ?

Black die ဘာျဖစျဖစ blue die 1 ကမည ျဖစရပ

blue dice black dice The set of favourable outcomes

1 1,2,3,4,5,6 ( 1,2 ),( 1,3 ),( 1,4 ),( 1,5 ),( 1,6 )


1 1,2,3,4,5,6 ( 1,1 ),( 1,2 ),( 1,3 ),( 1,4 ),( 1,5 ),( 1,6 )

( or )


6 1,2,3,4,5,6 ( 6,1 ),( 6,2 ),( 6,3 ),( 6,4 ),( 6,5 ),( 6,6 )


1 1,2,3,4,5,6 ( 1,1 ), ( 1,2 ),( 1,3 ),( 1,4 ),( 1,5 ),( 1,6 )

( or )


1,2,3,4,5,6 5 ( 1,5 ),( 2,5 ),( 3,5 ),( 4,5 ),( 5,5 ),( 6,5 )


1 5,6 ( 1,5 ),( 1,6 )



Number of possible outcomes = { ( 1,1 ), ( 1,2 ),( 1,3 ),( 1,4 ),( 1,5 ),( 1,6 ) }


P ( blue 1 ) = 6

36 =

1

6

---------------------------------------------------------------------

P ( black number > 4 ) = ?

blue dice ဘာျဖစျဖစ black dice 4 ထကႀကးရမည ျဖစရပ

Number of possible outcomes = { ( 1,6 ),( 2,6 ),( 3,6 ),( 4,6 ),( 5,6 ),( 6,6 )

( 1,5 ),( 2,5 ),( 3,5 ),( 4,5 ),( 5,5 ),( 6,5 ) }


P ( black number > 4 ) = 12

36 =

1

3

P ( blue 1 and black number > 4 ) = P ( blue 1 ) + P ( black number > 4 )

= 1

6 +

1

3 =

1

18

-----------------------------------------------------------------

The set of favourable outcomes Total Score

Even or

Odd

Total Score

Is Prime

No of Outcome

P ( A ) value

( 1,1 ) 2 E Prime 1 P ( 2 ) = 1

36

( 1,2 ),( 2,1 ) 3 O Prime 2 P ( 3 ) = 1

18

( 1,3 ),( 2,2 ),( 3,1 ) 4 E Not Prime 3 P ( 4 ) = 1

12

( 1,4 ),( 2,3 ),( 3,2 ),( 4,1 ) 5 O Prime 4 P ( 5 ) = 1

9

( 1,5 ),( 2,4 ),( 3,3 ),( 4,2 ), ( 5,1 ) 6 E Not Prime 5 P ( 6 ) = 5

36

( 1,6 ),( 2,5 ),( 3,4 ),( 4,3 ), ( 5,2 ), ( 6,1 ) 7 O Prime 6 P ( 7 ) = 1

6

( 2,6 ),( 3,5 ),( 4,4 ), ( 5,3 ), ( 6,2 ) 8 E Not Prime 5 P ( 8 ) = 5

36

( 3,6 ),( 4,5 ), ( 5,4 ), ( 6,3 ) 9 O Not Prime 4 P ( 9 ) = 1

9

( 4,6 ), ( 5,5 ), ( 6,4 ) 10 E Not Prime 3 P ( 10 ) = 1

12

( 5,6 ), ( 6,5 ) 11 O Prime 2 P ( 11 ) = 1

18

( 6,6 ) 12 E Not Prime 1 P ( 12 ) = 1

36

( 1 ) Probability of total score of 2 = P ( 2 ) = ?

The set of favourable outcomes = { ( 1,1 ) }


P ( 2 ) = 1

36

---------------------------------------------------------------------

P ( 3 ) = ?

The set of favourable outcomes = { ( 1 , 2 ) , ( 2 , 1 ) }


P ( 3 ) = 2

36 =

1

18

---------------------------------------------------------------------

P ( 4 ) = ?


1 1,2,3,4,5,6 ( 1,1 ), ( 1,2 ),( 1,3 ),( 1,4 ),( 1,5 ),( 1,6 )


1,2,3,4,5,6 5,6 ( 1,5 ),( 2,5 ),( 3,5 ),( 4,5 ),( 5,5 ),( 6,5 )

( 1,6 ),( 2,6 ),( 3,6 ),( 4,6 ),( 5,6 ),( 6,6 )



The set of favourable outcomes = { ( 1 , 3 ) , ( 2 , 2 ), ( 3 , 1 ) }


P ( 4 ) = 3

36 =

1

12

---------------------------------------------------------------------

P ( 5 ) = ?

The set of favourable outcomes = { ( 1 , 4 ) , ( 2 , 3 ), ( 3 , 2 ) , ( 4 , 1 ) }


P ( 5 ) = 4

36 =

1

9

---------------------------------------------------------------------

P ( 6 ) = ?

The set of favourable outcomes = { ( 1 , 5 ) , ( 2 , 4 ), ( 3 , 3 ) , ( 4 , 2 ) ,( 5 , 1 ) }


P ( 6 ) = 5

36

---------------------------------------------------------------------

P ( 7 ) = ?

The set of favourable outcomes = {( 1 , 6 ),( 2 , 5 ),( 3 , 4 ),( 4 , 3 ),( 5 , 2 ),( 6 , 1 )}


P ( 7 )

---------------------------------------------------------------------

P ( 8 ) = ?

The set of favourable outcomes = {( 2 , 6 ),( 3 , 5 ),( 4 , 4 ),( 5 , 3 ),( 6 , 2 ) }


P ( 8 ) = 5

36

---------------------------------------------------------------------

P ( 9 ) = ?

The set of favourable outcomes = { ( 3 , 6 ),( 4 , 5 ),( 5 , 4 ),( 6 , 3 ) }


P ( 9 ) = 4

36 =

1

9

P ( 10 ) = ?

The set of favourable outcomes = { ( 4 , 6 ),( 5 , 5 ),( 6 , 4 ) }


P ( 10 ) = 3

36 =

1

12

---------------------------------------------------------------------

P ( 11 ) = ?

The set of favourable outcomes = { ( 5 , 6 ),( 6 , 5 ) }


P ( 11 ) = 2

36 =

1

18

---------------------------------------------------------------------

P ( 12 ) = ?

The set of favourable outcomes = { ( 6 , 6 ) }


P ( 11 ) = 1

36

---------------------------------------------------------------------

Are all these outcomes are equally likely ?

အထကပါ outcome မားသည တနဖးတ outcome မားျဖစၾကပါသလား။

outcome တစခခငး၏ Probability တနဖးမား တမတက ေမးခငးျဖစသည။

No , they are not equally likely outcomes.

---------------------------------------------------------------------

( 2 ) the most likeiy score = ?

ႀကမေရအမားဆးျဖစေပၚေသာ ေပါငးလဒ တနဖး ( သ ) Probability တနဖးအမားဆး ရေစသည ေပါငးလဒ

the most likeiy score = 7

the least likeiy score = ?

ႀကမေရအနညးဆးျဖစေပၚေသာ ေပါငးလဒ တနဖး ( သ ) Probability တနဖးအနညးဆး ရေစသညေပါငးလဒ



the least likeiy score = 2 , 12

( 3 ) P ( total score is 2 or 12 ) = P ( 2 or 12 ) =?

P ( 2 or 12 ) = P ( 2 ) + P ( 12 ) = 1

36 +

1

36 =

2

36 =

1

18

2nd method

( စစေပါငးရလဒ 2 ( သ ) 12 ရေစမည ျဖစရပ = { ( 1 , 1 ) , ( 6 , 6 ) }

The set of favourable outcomes = { ( 1 , 1 ) , ( 6 , 6 ) }


P ( 2 or 12 ) = 2

36

ဟတြကလညး ရပါသည။ သ ေသာ အေပၚတြင ျဖစရပတစခခငးစအတြက Probability တနဖးမား

ရာျပသားရသျဖင ၄ငးတ အား အဆငသငသး၍ ခြရာေပးျခငးျဖစပါသည။ )

---------------------------------------------------------------------

( 4 ) P ( total score is 3 or 4 or 5 ) = P ( 3 or 4 or 5 ) =?

P ( 3 or 4 or 5 ) = P ( 3 ) + P ( 4 ) + P ( 5 ) = 2

36 +

3

36 +

4

36 =

6

36 =

1

4

---------------------------------------------------------------------

( 5 ) P ( total score is prime number ) = ? ( 2012 , ကခင ) ( 2013 , ရခင )

ႏစခေပါငးရလဒသည သဒၵကနးျဖစရမညဟဆသျဖင 2 ( သ ) 3 ( သ ) 5 ( သ ) 7 ( သ ) 11 တစခခ

ျဖစရပါမည။

P (total score is prime number ) = P ( 2 or 3 or 5 or 7 or 11 )

= P ( 2 ) + P ( 3 ) + P ( 5 ) + P ( 7 ) + P ( 11 ) = 1

36 +

3

36 +

4

36 +

6

36 +

2

36

= 15

36 =

5

12 ( 2014 , မြန၊ကရငတနသၤာရ )

---------------------------------------------------------------------

( 6 ) P ( total score is greater than 7 ) = P ( total score > 7 ) = ?

ႏစခေပါငးရလဒသည 7 ထကႀကးရမညဟဆသျဖင 8 ( သ ) 9 ( သ ) 10 ( သ ) 11 ( သ ) 12

တစခချဖစရပါမည။

P ( total score > 7 ) = P ( 8 or 9 or 10 or 11 or 12 )

= P ( 8 ) + P ( 9 ) + P ( 10 ) + P ( 11 ) + P ( 12 ) = 5

36 +

4

36 +

3

36 +

2

36 +

1

36

= 15

36 =

5

12

---------------------------------------------------------------------

( 8 ) P ( score on 2nd dice is greater than score on 1st dice ) =

P ( score on 2nd dice > score on 1st dice ) = ? ( 2014 , မြန၊ကရငတနသၤာရ )

The set of favourable outcomes = { ( 1,2 ),( 1,3 ),( 1,4 ),( 1,5 ),( 1,6 ),( 2,3 ),( 2,4 ),( 2,5 )

,( 2,6 ),( 3,4 ),( 3,5 ),( 3,6 ),( 4,5 ),( 4,6 ),( 5,6 ) }


P ( score on 2nd dice > score on 1st dice ) = 15

36 =

5

12

---------------------------------------------------------------------

1 mark

( 1 ) ( 2014 , Ygn ) ( 2007 , ပခး ) ( 2005 , မႏ ေလး )

poerfect square = ႏစထပကနးတမား

1 ၏ႏစထပကနးတ = 12 = 1 , 2 ၏ႏစထပကနးတ = 22

= 4 , 3 ၏ႏစထပကနးတ = 32 = 9

1st dice 2nd dice The set of favourable outcomes

1 2,3,4,5,6 ( 1,2 ),( 1,3 ),( 1,4 ),( 1,5 ),( 1,6 )

2 3,4,5,6 ( 2,3 ),( 2,4 ),( 2,5 ),( 2,6 )

3 4,5,6 ( 3,4 ),( 3,5 ),( 3,6 )

4 5,5 ( 4,5 ),( 4,6 )

5 6 ( 5,6 )



The set of favourable outcomes = { ( 1,3 ),( 2,2 ),( 3,1 ),( 3,6 ),( 4,5 ),( 5,4 ),( 6,3 ) }


P ( total score will be poerfect square ) = 7

36

---------------------------------------------------------------------

( 2 ) ( 2003 , Ygn )

P ( total score will not be poerfect square )

= 1 - P ( total score will be poerfect square )

= 1 - 7

36 =

29

36

---------------------------------------------------------------------

( 3 ) ( 2013 , ႏငငျခား )

P ( total score will be divisible by 4 ) = ?

The set of favourable outcomes = { ( 1,3 ),( 2,2 ),( 3,1 ),( 2,6 ),( 3,5 ),( 4,4 )

,( 5,3 ),( 6,2 ),( 6,6 ) }


P ( total score will be divisible by 4 ) = 9

36 =

1

4

---------------------------------------------------------------------

( 4 ) ( 2012 , Ygn )

P ( Blue 4 or Black 4 )

= P (( 4,1 ),( 4,2 ),( 4,3 ),( 4,4 ),( 4,5 ),( 4,6 ), ( 1,4 ),( 2,4 ),( 3,4 ),( 5,4 ),( 6,4 ))

= 11

36

---------------------------------------------------------------------

( 5 ) ( 2007 , စစကငး ။ခငး ) ( 2013 , Ygn )

P ( total score will be even number ) = 𝟏𝟖

𝟑𝟔 =

𝟏

𝟐

( 6 ) ( 2007 , ႏငငျခား )

The set of favourable outcomes = ( 3 , 2 ),( 6 , 2 ),( 3 , 4 ),( 6 , 4 ),( 3 , 6 ),( 6 , 6 )


P ( multiple of 3 on 1st die and multiple of 3 on 2nd die ) = 6

36 =

1

6

( or )

2nd Method

P ( multiple of 3 on 1st die and multiple of 3 on 2nd die )

= P ( 3 0r 6 on 1st die and 2 or 4 or 6 on 2nd die )

= 6

36 + 6

36 𝐱 6

36 + 6

36 + 6

36 = 1236 𝐱

1836 =

1

3 𝐱

1

2 = =

1

6

---------------------------------------------------------------------

perfect square The set of favourable outcomes ျဖစေနသညေပါငးလဒမား

4 ( 1,3 ), ( 2,2 ),( 3,1 ),

9 ( 3,6 ),( 4,5 ),( 5,4 ),( 6,3 )

4 ျဖငစား၍ျပတသည The set of favourable outcomes ျဖစေနသညေပါငးလဒမား

4 ( 1,3 ), ( 2,2 ),( 3,1 )

8 ( 2,6 ),( 3,5 ),( 5,3 ),( 6,2 )

12 ( 6,6 )

multiple of 3 on 1st die and multiple of 3 on 2nd die

1st dice 2nd dice The set of favourable outcomes

3,6 2,4,6 ( 3,2 ),( 3,2 ),( 3,6 ),( 6,2 ),( 6,4 ),( 6,6 )



( 8 ) total score = x , x < 7 , P ( x ) = = 1

9 =

4

36 ( 2012 , မြန၊ကရငတနသၤာရ )

The set of favourable outcomes Total Score

Total Score is Even or Odd

Total Score Is Prime

( 1,1 ) 2 E Prime

( 1,2 ),( 2,1 ) 3 O Prime

( 1,3 ),( 2,2 ),( 3,1 ) 4 E Not Prime

( 1,4 ),( 2,3 ),( 3,2 ),( 4,1 ) 5 O Prime

( 1,5 ),( 2,4 ),( 3,3 ),( 4,2 ), ( 5,1 ) 6 E Not Prime

( 1,6 ),( 2,5 ),( 3,4 ),( 4,3 ), ( 5,2 ), ( 6,1 ) 7 O Prime

( 2,6 ),( 3,5 ),( 4,4 ), ( 5,3 ), ( 6,2 ) 8 E Not Prime

( 3,6 ),( 4,5 ), ( 5,4 ), ( 6,3 ) 9 O Not Prime

( 4,6 ), ( 5,5 ), ( 6,4 ) 10 E Not Prime

( 5,6 ), ( 6,5 ) 11 O Prime

( 6,6 ) 12 E Not Prime

x = 5

---------------------------------------------------------------------

( 9 ) total score = x , x > 6 , P ( x ) = = 1

6 ( 2009 , ႏငငျခား )

x = ? Ans ;

---------------------------------------------------------------------

( 10 ) total score = x , x > 5 , P ( x ) = = 1

12 ( 2005 , ရမး၊ကယား )

x = ?

---------------------------------------------------------------------

( 11 ) P ( both die are even ) =

P ( 1st die is even and 2nd die is odd ) =

Old Question ( 5marks )

( 2014 , ရခင )

P ( total score is a multiple of 3 ) = ?

P ( the product of the score is divisible by 4 ) = ?

The set of all possible outcome = { ( 1 , 1 ), ( 1 , 2 ), ( 1 , 3 ),( 1 , 4 ),( 1 , 5 )

,( 1 , 6 ),( 2 , 1 ),( 2 , 2 ),( 2 , 3 ),( 2 , 4 ),( 2 , 5 ),( 2 , 6 ),( 3 , 1 ),( 3 , 2 )

,( 3 , 3 ),( 3 , 4 ),( 3 , 5 ),( 3 , 6 ),( 4 , 1 ),( 4 , 2 ),( 4 , 3 ),( 4 , 4 ),( 4 , 5 )

,( 4 , 6 ),( 5 , 1 ),( 5 , 2 ),( 5 , 3 ),( 5 , 4 ),( 5 , 5 ),( 5 , 6 ),( 6 , 1 ),( 6 , 2 )

,( 6 , 3 ),( 6 , 4 ),( 6 , 5 ),( 6 , 6 ) }




The set of favourable outcomes = { ( 1,2 ), ( 2,1 ),( 1,5 ),( 2,4 ),( 3,3 ),( 4,2 )

,( 5,1 ),( 3,6 ), ( 4,5 ),( 5,4 ),( 6,3 ),( 6,6 ) }


P ( total score is a multiple of 3 ) = 12

36 =

1

3

Multiple of 3 = 3,6,9,12

Multiple of 3 possible outcome ျဖစေနသညေပါငးလဒမား

3 ( 1,2 ), ( 2,1 )

6 ( 1,5 ),( 2,4 ),( 3,3 ),( 4,2 ) ,( 5,1 )

9 ( 3,6 ), ( 4,5 ),( 5,4 ),( 6,3 )

12 ( 6,6 )



Black die ( or ) 2nd die

Number of favourable outcomes = { ( 1 , 4 ),( 2 , 2 ),( 2 , 4 ),( 2 , 6 ),( 3 , 4 )

,( 4 , 1 ),( 4 , 2 ),( 4 , 3 ),( 4 , 4 ),( 4 , 5 )

,( 4 , 6 ),( 5 , 4 ),( 6 , 2 ),( 6 , 4 ),( 6 , 6 ) }

The set of favourable outcomes = 15

P ( the product of the score is divisible by 4 ) = = 15

36 =

5

12

( 2013 , မႏ ေလး )

( 1 ) table ဆြ

( 2 ) The set of all possible outcomes ေရး

( 3 ) Number of possible outcomes ေရး

P ( the sum of the scores is odd ) = ?

The set of favourable outcomes = { ( 1 , 2 ),( 1 , 4 ),( 1 , 6 ),( 2 , 1 ),( 2 , 3 )

,( 2 , 5 ),( 3 , 2 ),( 3 , 4 ),( 3 , 6 ),( 4 , 1 )

,( 4 , 3 ),( 4 , 5 ),( 5 , 2 ),( 5 , 4 ),( 5 , 6 )

,( 6 , 1 ),( 6 , 3 ),( 6 , 5 ) }


P ( the sum of the scores is odd ) = 18

36 =

1

2


P ( the product of the scores is greater than 15 ) = ?

The set of favourable outcomes = { ( 3,6 ),( 4,4 ),( 4,5 ),( 4,6 ) ,( 5,4 ),( 5,5 )

,( 5,6 ),( 6,3 ),( 6,4 ),( 6,5 ),( 6,6 ) }


P ( the product of the scores > 15 ) = = 11

36


P ( the product of the score is multiple of 6 ) = ?

Multiple of 6 6,12,18,24,30,36

The set of favourable outcomes = { ( 6 , 1 ),( 6 , 2 ),( 6 , 3 ),( 6 , 4 ),( 6 , 5 ),( 6 , 6 )

,( 2 , 3 ),( 2 , 6 ),( 3 , 2 ),( 3 , 4 ),( 3 , 6 ),( 4 , 3 )

,( 4 , 6 ),( 5 , 6 ),( 1 , 6 ) }


1 2 3 4 5 6

1

2

3

4

5

6

( 1 , 1 ) ( 1 , 2 ) ( 1 , 3 ) ( 1 , 4 ) ( 1 , 5 ) ( 1 , 6 )

( 2 , 1 ) ( 2 , 2 ) ( 2 , 3 ) ( 2 , 4 ) ( 2 , 5 ) ( 2 , 6 )

( 3 , 1 ) ( 3 , 2 ) ( 3 , 3 ) ( 3 , 4 ) ( 3 , 5 ) ( 3 , 6 )

( 4 , 1 ) ( 4 , 2 ) ( 4 , 3 ) ( 4 , 4 ) ( 4 , 5 ) ( 4 , 6 )

( 5 , 1 ) ( 5 , 2 ) ( 5 , 3 ) ( 5 , 4 ) ( 5 , 5 ) ( 5 , 6 )

( 6 , 1 ) ( 6 , 2 ) ( 6 , 3 ) ( 6 , 4 ) ( 6 , 5 ) ( 6 , 6 )

1 2 3 4 5 6

1

2

3

4

5

6

( 1 , 1 ) ( 1 , 2 ) ( 1 , 3 ) ( 1 , 4 ) ( 1 , 5 ) ( 1 , 6 )

( 2 , 1 ) ( 2 , 2 ) ( 2 , 3 ) ( 2 , 4 ) ( 2 , 5 ) ( 2 , 6 )

( 3 , 1 ) ( 3 , 2 ) ( 3 , 3 ) ( 3 , 4 ) ( 3 , 5 ) ( 3 , 6 )

( 4 , 1 ) ( 4 , 2 ) ( 4 , 3 ) ( 4 , 4 ) ( 4 , 5 ) ( 4 , 6 )

( 5 , 1 ) ( 5 , 2 ) ( 5 , 3 ) ( 5 , 4 ) ( 5 , 5 ) ( 5 , 6 )

( 6 , 1 ) ( 6 , 2 ) ( 6 , 3 ) ( 6 , 4 ) ( 6 , 5 ) ( 6 , 6 )

1 2 3 4 5 6

1

2

3

4

5

6

( 1 , 1 ) ( 1 , 2 ) ( 1 , 3 ) ( 1 , 4 ) ( 1 , 5 ) ( 1 , 6 )

( 2 , 1 ) ( 2 , 2 ) ( 2 , 3 ) ( 2 , 4 ) ( 2 , 5 ) ( 2 , 6 )

( 3 , 1 ) ( 3 , 2 ) ( 3 , 3 ) ( 3 , 4 ) ( 3 , 5 ) ( 3 , 6 )

( 4 , 1 ) ( 4 , 2 ) ( 4 , 3 ) ( 4 , 4 ) ( 4 , 5 ) ( 4 , 6 )

( 5 , 1 ) ( 5 , 2 ) ( 5 , 3 ) ( 5 , 4 ) ( 5 , 5 ) ( 5 , 6 )

( 6 , 1 ) ( 6 , 2 ) ( 6 , 3 ) ( 6 , 4 ) ( 6 , 5 ) ( 6 , 6 )

Blue die

Or

1st die

Blue die

Or

1st die

Blue die

Or

1st die



P ( the product of the score is multiple of 6 ) = = 15

36 =

5

12

---------------------------------------------------------------------

( 2013 , မြန၊ကရငတနသၤာရ )

P ( the sum of the scores is even ) = 1

2 ( ရာၿပး )

P ( the product of the scores is greater than 20 ) = = 1

6

P ( the product of the score is multiple of 6 ) = 5


( 2011 , Ygn )

P ( the sum of the scores is less than 7 ) = ?


Ans ; P ( the sum of the scores is less than 7 ) = 15

36 =

5

12

P ( the product of the scores is even ) = ?



,( 2 , 4 ),( 2 , 5 ),( 2 , 6 ),( 3 , 2 ),( 3 , 4 ),( 3 , 6 )

,( 4 , 1 ),( 4 , 2 ),( 4 , 3 ),( 4 , 4 ),( 4 , 5 ),( 4 , 6 )

,( 5 , 2 ),( 5 , 4 ),( 5 , 6 ),( 6 , 1 ),( 6 , 2 ),( 6 , 3 )

,( 6 , 4 ),( 6 , 5 ),( 6 , 6 ) }


P ( the product of the scores is even ) = 27

36 =

3

4

( 2012 , ရခင )

P ( score on 2nd die is greater than score on 1st die ) = = 5


**P ( score on one die is prime and score on other die is even ) = ?


The set of favourable outcomes = { ( 2 , 2 ) ( 2 , 3 ),( 2 , 4 ),( 2 , 6 ),( 2 , 5 ),( 3 , 2 )

,( 3 , 4 ),( 3 , 6 ),( 4 , 2 ),( 4 , 3 ),( 4 , 5 ),( 5 , 2 )

1 2 3 4 5 6

1

2

3

4

5

6

( 1 , 1 ) ( 1 , 2 ) ( 1 , 3 ) ( 1 , 4 ) ( 1 , 5 ) ( 1 , 6 )

( 2 , 1 ) ( 2 , 2 ) ( 2 , 3 ) ( 2 , 4 ) ( 2 , 5 ) ( 2 , 6 )

( 3 , 1 ) ( 3 , 2 ) ( 3 , 3 ) ( 3 , 4 ) ( 3 , 5 ) ( 3 , 6 )

( 4 , 1 ) ( 4 , 2 ) ( 4 , 3 ) ( 4 , 4 ) ( 4 , 5 ) ( 4 , 6 )

( 5 , 1 ) ( 5 , 2 ) ( 5 , 3 ) ( 5 , 4 ) ( 5 , 5 ) ( 5 , 6 )

( 6 , 1 ) ( 6 , 2 ) ( 6 , 3 ) ( 6 , 4 ) ( 6 , 5 ) ( 6 , 6 )

1 2 3 4 5 6

1

2

3

4

5

6

( 1 , 1 ) ( 1 , 2 ) ( 1 , 3 ) ( 1 , 4 ) ( 1 , 5 ) ( 1 , 6 )

( 2 , 1 ) ( 2 , 2 ) ( 2 , 3 ) ( 2 , 4 ) ( 2 , 5 ) ( 2 , 6 )

( 3 , 1 ) ( 3 , 2 ) ( 3 , 3 ) ( 3 , 4 ) ( 3 , 5 ) ( 3 , 6 )

( 4 , 1 ) ( 4 , 2 ) ( 4 , 3 ) ( 4 , 4 ) ( 4 , 5 ) ( 4 , 6 )

( 5 , 1 ) ( 5 , 2 ) ( 5 , 3 ) ( 5 , 4 ) ( 5 , 5 ) ( 5 , 6 )

( 6 , 1 ) ( 6 , 2 ) ( 6 , 3 ) ( 6 , 4 ) ( 6 , 5 ) ( 6 , 6 )

1 2 3 4 5 6

1

2

3

4

5

6

( 1 , 1 ) ( 1 , 2 ) ( 1 , 3 ) ( 1 , 4 ) ( 1 , 5 ) ( 1 , 6 )

( 2 , 1 ) ( 2 , 2 ) ( 2 , 3 ) ( 2 , 4 ) ( 2 , 5 ) ( 2 , 6 )

( 3 , 1 ) ( 3 , 2 ) ( 3 , 3 ) ( 3 , 4 ) ( 3 , 5 ) ( 3 , 6 )

( 4 , 1 ) ( 4 , 2 ) ( 4 , 3 ) ( 4 , 4 ) ( 4 , 5 ) ( 4 , 6 )

( 5 , 1 ) ( 5 , 2 ) ( 5 , 3 ) ( 5 , 4 ) ( 5 , 5 ) ( 5 , 6 )

( 6 , 1 ) ( 6 , 2 ) ( 6 , 3 ) ( 6 , 4 ) ( 6 , 5 ) ( 6 , 6 )

Blue die

Or

1st die

Blue die

Or

1st die

အထးသတထားရန

Event တနဖးမားရာရာတြင sum ႏင product က အထးသတထားပါ။

ပစာၦက sum ကရာခငးတာလား product ကရာခငးတာလား အထးဂရစကရနလပါသည။

Blue die

Or

1st die



,( 5 , 4 ),( 5 , 6 ),( 6 , 5 ),( 6 , 3 ),( 6 , 2 ) }


**P ( score on one die is prime and score on other die is even ) = = 17

36

( 2010 , ကခင )

P ( total score is a multiple of 3 ) = 12

36 =

1


P ( the product of the score is divisible by 4 ) = 15

36 =

5


( 2010 , ရမး၊ကယား )

P ( total score is prime number ) = 15

36 =

5


P ( the total score is greater than 10 ) = ?


The set of favourable outcomes = { ( 6 , 6 ),( 6 , 5 ),( 5 , 6 ) }


P ( the total score is greater than 10 ) = = 3

36 =

1

12

( 2010 , ႏငငျခား )

P ( getting a total of 10 or more


The set of favourable outcomes = { ( 4,6 ),( 5,5 ),( 5,6 ),( 6,4 ),( 6,5 ),( 6,6 )}


P ( getting a total of 10 or more ) = = 6

36 =

1

6

P ( both dice show the same number ) = ?

The set of favourable outcomes = { ( 1,1 ),( 2,2 ),( 3,3 ),( 4,4 ),( 5,5 ),( 6,6 ) }


P ( both dice show the same number ) = = 6

36 =

1

6

( 2009 , မႏ ေလး )

P ( the sum of the score is prime number ) = P ( total score is prime number ) = 5

12

( ရာၿပး )

P ( the product of the score is divisible by 6 or 9 ) = ?


1 2 3 4 5 6

1

2

3

4

5

6

( 1 , 1 ) ( 1 , 2 ) ( 1 , 3 ) ( 1 , 4 ) ( 1 , 5 ) ( 1 , 6 )

( 2 , 1 ) ( 2 , 2 ) ( 2 , 3 ) ( 2 , 4 ) ( 2 , 5 ) ( 2 , 6 )

( 3 , 1 ) ( 3 , 2 ) ( 3 , 3 ) ( 3 , 4 ) ( 3 , 5 ) ( 3 , 6 )

( 4 , 1 ) ( 4 , 2 ) ( 4 , 3 ) ( 4 , 4 ) ( 4 , 5 ) ( 4 , 6 )

( 5 , 1 ) ( 5 , 2 ) ( 5 , 3 ) ( 5 , 4 ) ( 5 , 5 ) ( 5 , 6 )

( 6 , 1 ) ( 6 , 2 ) ( 6 , 3 ) ( 6 , 4 ) ( 6 , 5 ) ( 6 , 6 )

1 2 3 4 5 6

1

2

3

4

5

6

( 1 , 1 ) ( 1 , 2 ) ( 1 , 3 ) ( 1 , 4 ) ( 1 , 5 ) ( 1 , 6 )

( 2 , 1 ) ( 2 , 2 ) ( 2 , 3 ) ( 2 , 4 ) ( 2 , 5 ) ( 2 , 6 )

( 3 , 1 ) ( 3 , 2 ) ( 3 , 3 ) ( 3 , 4 ) ( 3 , 5 ) ( 3 , 6 )

( 4 , 1 ) ( 4 , 2 ) ( 4 , 3 ) ( 4 , 4 ) ( 4 , 5 ) ( 4 , 6 )

( 5 , 1 ) ( 5 , 2 ) ( 5 , 3 ) ( 5 , 4 ) ( 5 , 5 ) ( 5 , 6 )

( 6 , 1 ) ( 6 , 2 ) ( 6 , 3 ) ( 6 , 4 ) ( 6 , 5 ) ( 6 , 6 )

Blue die

Or

1st die

Blue die

Or

1st die



The set of favourable outcomes = { ( 1 , 6 ),( 2 , 3 ),( 2 , 6 ),( 6 , 1 ).( 6 , 2 ),( 6 , 3 )

( 6 , 4 ),( 6 , 5 ),( 6 , 6 ),( 3 , 6 ),( 4 , 3 ),( 4 , 6 )

( 5 , 6 ),( 3 , 2 ),( 3 , 3 ),( 3 , 4 ) }


P ( the product of the score is divisible by 6 or 9 ) = 16

36 =

4

9

( 2009 , စစကငး၊ခငး )

P ( total score is less than 7 ) = P ( total score < 7 ) = 5

12

P ( the total score is not divisible by 3 ) = ?



,( 2 , 4 ),( 2 , 5 ),( 4 , 1 ),( 4 , 2 ),( 4 , 4 ),( 4 , 5 )

,( 5 , 1 ),( 5 , 2 ),( 5 , 4 ),( 5 , 5 ) }


P ( the total score is not divisible by 3 ) = = 16

36 =

4

9

( 2009 , ရခင )

P ( total score is greater than 5 ) = P ( total score > 5 ) = = 13

18

P ( the total score is divisible by ) = = 18

36 =

1

2

( 2006 , ပခး )

P ( score on 1st die is 2 less than score on 2nd die ) = ?

The set of favourable outcomes = { ( 1,3 ),( 2,4 ),( 3,5 ),( 4,6 ) }


P ( score on 1st die is 2 less than score on 2nd die ) = 4

36 =

1

9

( 2008 , ရမး၊ကယား )

In a game , two dice ( dice A and B ) are used. Die A has 2 blue faces and 4 white

faces. Die B has 4 blue faces and 2 red faces. Die A and B are throw together. Find the

probability that just one die show a blue face on top.

Let B1 , B

2 ,

W1 , W

2 , W

3

, W4 are six

faces of die A

And b1 , b

2 ,

b3 , b

4 , r

1 , r

2 are six faces of die B

1 2 3 4 5 6

1

2

3

4

5

6

( 1 , 1 ) ( 1 , 2 ) ( 1 , 3 ) ( 1 , 4 ) ( 1 , 5 ) ( 1 , 6 )

( 2 , 1 ) ( 2 , 2 ) ( 2 , 3 ) ( 2 , 4 ) ( 2 , 5 ) ( 2 , 6 )

( 3 , 1 ) ( 3 , 2 ) ( 3 , 3 ) ( 3 , 4 ) ( 3 , 5 ) ( 3 , 6 )

( 4 , 1 ) ( 4 , 2 ) ( 4 , 3 ) ( 4 , 4 ) ( 4 , 5 ) ( 4 , 6 )

( 5 , 1 ) ( 5 , 2 ) ( 5 , 3 ) ( 5 , 4 ) ( 5 , 5 ) ( 5 , 6 )

( 6 , 1 ) ( 6 , 2 ) ( 6 , 3 ) ( 6 , 4 ) ( 6 , 5 ) ( 6 , 6 )

1 2 3 4 5 6

1

2

3

4

5

6

( 1 , 1 ) ( 1 , 2 ) ( 1 , 3 ) ( 1 , 4 ) ( 1 , 5 ) ( 1 , 6 )

( 2 , 1 ) ( 2 , 2 ) ( 2 , 3 ) ( 2 , 4 ) ( 2 , 5 ) ( 2 , 6 )

( 3 , 1 ) ( 3 , 2 ) ( 3 , 3 ) ( 3 , 4 ) ( 3 , 5 ) ( 3 , 6 )

( 4 , 1 ) ( 4 , 2 ) ( 4 , 3 ) ( 4 , 4 ) ( 4 , 5 ) ( 4 , 6 )

( 5 , 1 ) ( 5 , 2 ) ( 5 , 3 ) ( 5 , 4 ) ( 5 , 5 ) ( 5 , 6 )

( 6 , 1 ) ( 6 , 2 ) ( 6 , 3 ) ( 6 , 4 ) ( 6 , 5 ) ( 6 , 6 )

Blue die

Or

1st die

Blue die

Or

1st die



Die B

Just = exactly

just one die show a blue face on top = exactly one die show a blue face on top

The set of all possible outcome = { ( B1,b

1 ),( B

1,b

2 ),( B

1,r

1 ),( B

1,r

2 ),( B

1,r

3 )

,( B1,r

4 ),( B

2,b

1 ),( B

2 ,b

2 ),( B

2,r

1 ),( B

2,r

2 ),( B

2,r

3 ),( B

2,r

4 ),( B

3,b

1 )

,( B3

,b2 ),( B

3,r

1 ),( B

3,r

2 ),( B

3,r

3 ),( B

3,r

4 ),( B

4,b

1 ),( B

4,b

2 ),( B

4,r

1 )

,( B4,r

2 ),( B

4,r

3 ),( B

4,r

4 ),( W

1,b

1 ),( W

1,b

2 ),( W

1,r

1 ),( W

1,r

2 ),( W

1,r

3 )

,( W1,r

4 )m( W

2,b

1 ),( W

2,b

2 ),( W

2,r

1 ),( W

2,r

2 ),( W

2,r

3 ),( W

2,r

4 ) }


Both dice show blue.

Just one die show blue.

P ( just one die show a blue face on top ) = ?

The set of favourable outcomes = { ( B1,r

1 ),( B

1,r

2 ),( B

1,r

3 ),( B

1,r

4 ),( B

2,r

1 )

,( B2,r

2 ),( B

2,r

3 ),( B

2,r

4 ),( B

3,r

1 ),( B

3,r

2 )

,( B3,r

3 ),( B

3,r

4 ),( B

4,r

1 ),( B

4,r

2 ),( B

4,r

3 )

,( B4,r

4 ),( W

1,b

1 ),( W

1,b

2 ),( W

2,b

1 ),( W

2,b

2 )


P ( just one die show a blue face on top ) = 20

36 =

5

9

( or )

P ( just one die show a blue face on top ) = ?

P ( blue face on die A ) = 2

6 =

1

3

P ( white face on die A ) = P ( not blue face on die A ) = 4

6 =

2

3

P ( blue face on die B ) = 4

6 =

2

3

P ( white face on die B ) = P ( not blue face on die B ) = 2

6 =

1

3

P ( just one die show a blue face on top )

= P ( blue face on die A and not blue face on die B ( or ) not blue face on die A and

blue face on die B )

= P ( blue face on die A and not blue face on die B ) + P ( not blue face on die A and

blue face on die B )

= [ P ( blue face on die A ) x P ( not blue face on die B ) ]

+ [ P ( not blue face on die A ) x P ( blue face on die B ) ]

= [ 1

3 x

1

3 ] + [

2

3 x

2

3 ]

= 1

9 +

4

9

= 5

9

b1 b2 r1 r2 r3 r4

B1

B2

B3

B4

W1

W2

( B1,b1 ) ( B1,b2 ) ( B1,r1 ) ( B1,r2 ) ( B1,r3 ) ( B1,r4 )

( B2,b1 ) ( B2 ,b2 ) ( B2,r1 ) ( B2,r2 ) ( B2,r3 ) ( B2,r4 )

( B3,b1 ) ( B3 ,b2 ) ( B3,r1 ) ( B3,r2 ) ( B3,r3 ) ( B3,r4 )

( B4,b1 ) ( B4,b2 ) ( B4,r1 ) ( B4,r2 ) ( B4,r3 ) ( B4,r4 )

( W1,b1 ) ( W1,b2 ) ( W1,r1 ) ( W1,r2 ) ( W1,r3 ) ( W1,r4 )

( W2,b1 ) ( W2,b2 ) ( W2,r1 ) ( W2,r2 ) ( W2,r3 ) ( W2,r4 )

Die A



ေနာကဆကတြေမးခြနးမား

( a )

P ( neither dice show a blue face on top ) ( သ႕မဟတ )

P ( neither die A nor Die B show a blue face on top ) = ?

P ( neither dice show a blue face on top )

= P ( both dice don’t show a blue face on top )

= P ( not blue face on die A and not blue face on die B )

= P ( not blue face on die A ) x P ( not blue face on die B )

= 2

3 x

1

3 =

2

9

အေပၚရ ဇယားကြကမတြကလင ၄ငးအေျခအေနမာ

P ( ( W1,r1 ),( W1,r2 ),( W1,r3 ),( W1,r4 ),( W1,r1 ),( W1,r2 ),( W1,r3 ),( W1,r4 ) ) = 8

36 =

2

9

( b )

P ( either die A or Die B show a blue face on top ) = ?

Since throwing two dice are mutually exclusive and indepewndent event

P ( either die A or Die B show a blue face on top )

= P ( just one die show a blue face on top ) = 20

36 =

5

9

( c )

P ( at least one die show a blue face on top ) = ?

P ( at least one die show a blue face on top )


blue face on die B ( or ) blue face on die A and blue face on die B)



+ [ P ( blue face on die A ) x P ( blue face on die B ) ]

= [ 1

3 x

1

3 ] + [

2

3 x

2

3 ] + [

1

3 x

2

3 ]

= 1

9 +

4

9 +

2

9

= 7

9

အေပၚရ ဇယားကြကမတြကလင ၄ငးအေျခအေနမာ

P (( B1,r1 ),( B1,r2 ),( B1,r3 ) ,( B1,r4 ),( B2,b1 ),( B2 ,b2 ),( B2,r1 ),( B2,r2 ),( B2,r3 ),

( B2,r4 ),( B3,r1 ),( B3,r2 ),( B3,r3 ),( B3,r4 ),( B4,r1 ),( B4,r2 ),( B4,r3 ),( B4,r4 )

,( W1,b1 ),( W1,b2 ),( W1,r1 ),( W1,r2 ),( W1,r3 ),( W1,r4 )m( W2,b1 ),( W2,b2 ),( W2,r1 ),( W2,r2 )

,( W2,r3 ),( W2,r4 ) ) = 28

36 =

7

9

( d )

P ( at most one die show a blue face on top ) = ?

P ( at most one die show a blue face on top )


blue face on die B ( or ) not blue face on die A and not blue face on die B)



+ [ P ( blue face on die A ) x P ( blue face on die B ) ]

= [ 1

3 x

1

3 ] + [

2

3 x

2

3 ] + [

1

3 x

2

3 ]

= 1

9 +

4

9 +

2

9

= 7

9

---------------------------------------------------------------------



Table အတြက ေနာကဆကတြေမးခြနးမား

( 1 )

P ( at least one die show 3 ) = ?


P ( at least one die show 3 ) = 11

36

---------------------------------------------------------------------

( 2 )

P ( at most one die show 3 ) = ?


P ( at most one die show 3 ) = 35

36

---------------------------------------------------------------------

( 3 )P ( just one die show 3 ) = 10

36


---------------------------------------------------------------------

( 4 )P (either die A or die B show 3 ) = 10

36


1 2 3 4 5 6

1

2

3

4

5

6

( 1 , 1 ) ( 1 , 2 ) ( 1 , 3 ) ( 1 , 4 ) ( 1 , 5 ) ( 1 , 6 )

( 2 , 1 ) ( 2 , 2 ) ( 2 , 3 ) ( 2 , 4 ) ( 2 , 5 ) ( 2 , 6 )

( 3 , 1 ) ( 3 , 2 ) ( 3 , 3 ) ( 3 , 4 ) ( 3 , 5 ) ( 3 , 6 )

( 4 , 1 ) ( 4 , 2 ) ( 4 , 3 ) ( 4 , 4 ) ( 4 , 5 ) ( 4 , 6 )

( 5 , 1 ) ( 5 , 2 ) ( 5 , 3 ) ( 5 , 4 ) ( 5 , 5 ) ( 5 , 6 )

( 6 , 1 ) ( 6 , 2 ) ( 6 , 3 ) ( 6 , 4 ) ( 6 , 5 ) ( 6 , 6 )

1 2 3 4 5 6

1

2

3

4

5

6

( 1 , 1 ) ( 1 , 2 ) ( 1 , 3 ) ( 1 , 4 ) ( 1 , 5 ) ( 1 , 6 )

( 2 , 1 ) ( 2 , 2 ) ( 2 , 3 ) ( 2 , 4 ) ( 2 , 5 ) ( 2 , 6 )

( 3 , 1 ) ( 3 , 2 ) ( 3 , 3 ) ( 3 , 4 ) ( 3 , 5 ) ( 3 , 6 )

( 4 , 1 ) ( 4 , 2 ) ( 4 , 3 ) ( 4 , 4 ) ( 4 , 5 ) ( 4 , 6 )

( 5 , 1 ) ( 5 , 2 ) ( 5 , 3 ) ( 5 , 4 ) ( 5 , 5 ) ( 5 , 6 )

( 6 , 1 ) ( 6 , 2 ) ( 6 , 3 ) ( 6 , 4 ) ( 6 , 5 ) ( 6 , 6 )

1 2 3 4 5 6

1

2

3

4

5

6

( 1 , 1 ) ( 1 , 2 ) ( 1 , 3 ) ( 1 , 4 ) ( 1 , 5 ) ( 1 , 6 )

( 2 , 1 ) ( 2 , 2 ) ( 2 , 3 ) ( 2 , 4 ) ( 2 , 5 ) ( 2 , 6 )

( 3 , 1 ) ( 3 , 2 ) ( 3 , 3 ) ( 3 , 4 ) ( 3 , 5 ) ( 3 , 6 )

( 4 , 1 ) ( 4 , 2 ) ( 4 , 3 ) ( 4 , 4 ) ( 4 , 5 ) ( 4 , 6 )

( 5 , 1 ) ( 5 , 2 ) ( 5 , 3 ) ( 5 , 4 ) ( 5 , 5 ) ( 5 , 6 )

( 6 , 1 ) ( 6 , 2 ) ( 6 , 3 ) ( 6 , 4 ) ( 6 , 5 ) ( 6 , 6 )

1 2 3 4 5 6

1

2

3

4

5

6

( 1 , 1 ) ( 1 , 2 ) ( 1 , 3 ) ( 1 , 4 ) ( 1 , 5 ) ( 1 , 6 )

( 2 , 1 ) ( 2 , 2 ) ( 2 , 3 ) ( 2 , 4 ) ( 2 , 5 ) ( 2 , 6 )

( 3 , 1 ) ( 3 , 2 ) ( 3 , 3 ) ( 3 , 4 ) ( 3 , 5 ) ( 3 , 6 )

( 4 , 1 ) ( 4 , 2 ) ( 4 , 3 ) ( 4 , 4 ) ( 4 , 5 ) ( 4 , 6 )

( 5 , 1 ) ( 5 , 2 ) ( 5 , 3 ) ( 5 , 4 ) ( 5 , 5 ) ( 5 , 6 )

( 6 , 1 ) ( 6 , 2 ) ( 6 , 3 ) ( 6 , 4 ) ( 6 , 5 ) ( 6 , 6 )

Blue die

Or

1st die

Blue die

Or

1st die

Blue die

Or

1st die

Blue die

Or

1st die



( 5 ) P ( exactly one show 3 ) = 10

36


---------------------------------------------------------------------

( 6 )

P ( neither die A nor die B show 3 ) = ?


P ( neither die A nor die B show 3 ) = 25

36

---------------------------------------------------------------------

( 7 )

P ( both dice show difference numbers ) = ?


P ( both dice show difference numbers )

= 1 - P ( both dice show same numbers ) = 1 - 6

36 =

30

36 =

5

6

---------------------------------------------------------------------

---------------------------------------------------------------------

1 2 3 4 5 6

1

2

3

4

5

6

( 1 , 1 ) ( 1 , 2 ) ( 1 , 3 ) ( 1 , 4 ) ( 1 , 5 ) ( 1 , 6 )

( 2 , 1 ) ( 2 , 2 ) ( 2 , 3 ) ( 2 , 4 ) ( 2 , 5 ) ( 2 , 6 )

( 3 , 1 ) ( 3 , 2 ) ( 3 , 3 ) ( 3 , 4 ) ( 3 , 5 ) ( 3 , 6 )

( 4 , 1 ) ( 4 , 2 ) ( 4 , 3 ) ( 4 , 4 ) ( 4 , 5 ) ( 4 , 6 )

( 5 , 1 ) ( 5 , 2 ) ( 5 , 3 ) ( 5 , 4 ) ( 5 , 5 ) ( 5 , 6 )

( 6 , 1 ) ( 6 , 2 ) ( 6 , 3 ) ( 6 , 4 ) ( 6 , 5 ) ( 6 , 6 )

1 2 3 4 5 6

1

2

3

4

5

6

( 1 , 1 ) ( 1 , 2 ) ( 1 , 3 ) ( 1 , 4 ) ( 1 , 5 ) ( 1 , 6 )

( 2 , 1 ) ( 2 , 2 ) ( 2 , 3 ) ( 2 , 4 ) ( 2 , 5 ) ( 2 , 6 )

( 3 , 1 ) ( 3 , 2 ) ( 3 , 3 ) ( 3 , 4 ) ( 3 , 5 ) ( 3 , 6 )

( 4 , 1 ) ( 4 , 2 ) ( 4 , 3 ) ( 4 , 4 ) ( 4 , 5 ) ( 4 , 6 )

( 5 , 1 ) ( 5 , 2 ) ( 5 , 3 ) ( 5 , 4 ) ( 5 , 5 ) ( 5 , 6 )

( 6 , 1 ) ( 6 , 2 ) ( 6 , 3 ) ( 6 , 4 ) ( 6 , 5 ) ( 6 , 6 )

1 2 3 4 5 6

1

2

3

4

5

6

( 1 , 1 ) ( 1 , 2 ) ( 1 , 3 ) ( 1 , 4 ) ( 1 , 5 ) ( 1 , 6 )

( 2 , 1 ) ( 2 , 2 ) ( 2 , 3 ) ( 2 , 4 ) ( 2 , 5 ) ( 2 , 6 )

( 3 , 1 ) ( 3 , 2 ) ( 3 , 3 ) ( 3 , 4 ) ( 3 , 5 ) ( 3 , 6 )

( 4 , 1 ) ( 4 , 2 ) ( 4 , 3 ) ( 4 , 4 ) ( 4 , 5 ) ( 4 , 6 )

( 5 , 1 ) ( 5 , 2 ) ( 5 , 3 ) ( 5 , 4 ) ( 5 , 5 ) ( 5 , 6 )

( 6 , 1 ) ( 6 , 2 ) ( 6 , 3 ) ( 6 , 4 ) ( 6 , 5 ) ( 6 , 6 )

ေပါငးလဒမား 2 3 4 5 6 7 8 9 10 11 12

( 1st , 2nd ) (blue,black )

( 1,1 ) ( 1,2 ) ( 1,3 ) ( 1,4 ) ( 1,5 ) ( 1,6 )

( 2,1 ) ( 2,2 ) ( 2,3 ) ( 2,4 ) ( 2,5 ) ( 2,6 )

( 3,1 ) ( 3,2 ) ( 3,3 ) ( 3,4 ) ( 3,5 ) ( 3,6 )

( 4,1 ) ( 4,2 ) ( 4,3 ) ( 4,4 ) ( 4,5 ) ( 4,6 )

( 5,1 ) ( 5,2 ) ( 5,3 ) ( 5,4 ) ( 5,5 ) ( 5,6 )

( 6,1 ) ( 6,2 ) ( 6,3 ) ( 6,4 ) ( 6,5 ) ( 6,6 )

Blue die

Or

1st die

Blue die

Or

1st die

Blue die

Or

1st die



Tossing Coin Problems

Exercise ( 7.2 )

( 11 ) Copy and complete the table for the toss of a coin and the roll of a die.

2nd Coin

The set of all possible outcome = { ( H , H ),( H , T ),( T , H ),( T , T ) }


P ( H , H ) = P ( getting two heads ) = ? ( 2012 , ရနကန )

The set of favourable outcomes = { ( H , H ) }


P ( H , H ) = 1

4

P ( T , T ) = ?

The set of favourable outcomes = { ( T , T ) }


P ( T , T ) = 1

4

P ( a head and a tail in any order ) = ?

The set of favourable outcomes = { ( H , T ) , ( T , H ) }


P ( T , T ) = 2

4 =

1

2

ေနာကဆကတြေလကငရန

P ( at least one tail ) = ? ( 2015 , ျပညတြငး )

The set of favourable outcomes ={ ( H , H ), ( H , T ),( T , H ),( T , T ) }


P ( at least one tail ) = 3

4

---------------------------------------------------------------------

P ( at least one head ) = ? ( 2011 , ရနကန ) ( 2009 , ရနကန၊ မႏ ေလး )

The set of favourable outcomes = { ( H , H ),( H , T ),( T , H ) ,( T , T ) }


P ( at least one head ) = 3

4

---------------------------------------------------------------------

P ( at most one tail ) = ? ( 2013 , ပခး )

The set of favourable outcomes = { ( H , H ),( H , T ),( T , H ), ( T , T ) }


P ( at most one tail ) = 3

4

---------------------------------------------------------------------

P ( not getting two tails ) = ?

The set of favourable outcomes = { ( H , H ),( H , T ),( T , H ), ( T , T ) }


P (not getting two tails ) = 3

4

---------------------------------------------------------------------

P ( at most one head ) = ? ( 2010 , ႏငငျခား )

The set of favourable outcomes = { ( H ,

H ) ,( H , T ),( T , H ),( T , T ) }

H T

H ( H , H ) ( H , T )

T ( T , H ) ( T , T )

1st Coin



Number of favourable outcomes =

P ( ( H , T ),( T , H ),( T , T ) ) = 3

4

---------------------------------------------------------------------

P ( not getting two heads ) = ?

The set of favourable outcomes = { ( H , H ) ,( H , T ),( T , H ),( T , T ) }


P ( not getting two heads ) = 3

4

---------------------------------------------------------------------

P ( exactly one head ) = P ({ ( H , H ), ( H , T ),( T , H ) ,( T , T ) } )

P ( just one head ) = P ({ ( H , H ), ( H , T ),( T , H ) ,( T , T ) } )

P ( exactly one tail ) = P ({ ( H , H ), ( H , T ),( T , H ) ,( T , T ) } )

P ( just one tail ) = P ({ ( H , H ), ( H , T ),( T , H ) ,( T , T ) } )

P ( ( H , T ),( T , H ) ) = 3

4 ( 2010 , ရမး၊ကယား )

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( 15 ) Copy and complete this array of any ordered triples for the possible outcomes

When 3 coins are tossed simultaneously;

( a ) P ( exactly two Heads ) = ?

The set of all possible outcome = { HHH, HHT, HTH, HTT, THH, THT, TTH, TTT }


The set of favourable outcomes ={ HHT, HTH, THH }


8

( b ) P ( two Heads and a Tail in any order ) = ?

The set of favourable outcomes ={ HHT, HTH, THH }


8

( c ) P ( 3 Tails ) = ?

The set of favourable outcomes = { TTT }


P ( 3 Tails ) = 1

---------------------------------------------------------------------


P ( exactly one Head ) = ?

The set of favourable outcomes = { HHH, HHT, HTH, HTT, THH, THT, TTH, TTT }


P ( exactly one Head ) = 3

8

---------------------------------------------------------------------

P ( exactly one Tail ) = ?



P ( exactly one Tail ) = 3

8

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P ( at least one tail ) = ?




8

---------------------------------------------------------------------

P ( at least one head ) = ? ( 2013 , ကခင )



HHH HHT HTH HTT

THH THT TTH TTT




8

---------------------------------------------------------------------

P ( at most one tail ) = ?




8 =

1

2

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P ( at most one head ) = ?



P ( at most one head ) = 4

8 =

1

2

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P ( just one head ) = P ( exactly one head )

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P ( just one tail ) = P ( exactly one tail )

---------------------------------------------------------------------

P ( exactly two Heads ) = ?




8

---------------------------------------------------------------------

P ( exactly two Tails ) = ? ( 2011 , မြနကရင၊တနသၤာရ )




8

---------------------------------------------------------------------

P ( at least two tails ) = ?




8 =

1

2

---------------------------------------------------------------------

P ( at least two heads ) = ? ( 2014 , ကခင )


Number of favourable outcomes =


8 =

1

2

---------------------------------------------------------------------

P ( at most two tails ) = ?



P ( at most two tails ) = 7

8

---------------------------------------------------------------------

P ( at most two heads ) = ?



P ( at most one head ) = 7

8

---------------------------------------------------------------------

P ( just two heads ) = P ( exactly two heads )

---------------------------------------------------------------------

P ( just two tails ) = P ( exactly two tails )

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( 2016 , ႏငငျခား ) ( 2011 , မႏ ေလး မာ 6 ႀကမေမး )

Number of possible outcomes for tossing 5 fair coils = 2 x 2 x 2 x 2 x 2 = 32

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( 2014 , ပခး )

Number of possible outcomes for tossing a fair coils 3 times and rolling a die once



= 2 x 2 x 2 x 6 = 48

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( 2017 , ျပညတြငး )

A coin is toss 2 times ,

P ( at least one tail ) = x – 2



x – 2 = 3

4

x = 3

4 + 2

x = 3+8

4 =

11

4

( 2011 , ပခး )

3 coin are toss ,

P ( at least one head ) = 1 – x

x = ? Ans ; 1

8

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( 2008 , ႏငငျခား )

3 coin are toss ,

P ( exactly two tails and a head ) = ?



P ( exactly two tails and a head ) = 3

8

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5 marks

( 2012 , ကခင )

A coin is toss 4 times. Head or Tail is record each time. Draw a tree diagram..Find the

probabilities of exactly one tail and at least one tail .

1st Toss 2nd Toss 3rd Toss 4th Toss Possible Outcomes

H ( H , H , H , H )

T ( H , H , H , T )

H ( H , H , T , H )

T ( H , H , T , T )

H ( H , T , H , H )

T ( H , T , H , T )

H ( H , T , T , H )

T ( H , T , T , T )

H ( T , H , H , H )

T ( T , H , H , T )

H ( T , H , T , H )

T ( T , H , T , T )

H ( T , T , H , H )

T ( T , T , H , T )

H ( T , T , T , H )

T ( T , T , T , T )

The set of all possible outcome = { ( H,H,H,H ),( H,H,H,T ), ( H,H,T,H ),( H,H,T,T )

, ( H,T,H,H ),( H,T,H,T ),( H,T,T,H ),( H,T,T,T ),( T , H , H , H ),( T , H , H , T )

,( T , H , T , H ),( T , H , T , T ),( T , T , H , H ),( T , T , H , T ),( T , T , T , H )

,( T , T , T , T ) }


H

T

H

T

H

T

H

H

T

H

T

H

T

T



P ( exactly one tail ) = ?

The set of favourable outcomes ={ ( H,H,H,H ),( H,H,H,T ), ( H,H,T,H ),( H,H,T,T )



,( T , T , T , T ) }


P ( exactly one tail ) = 3

16

P ( at least one tail ) = ?




,( T , T , T , T ) }



16

( 2007 , ရမး၊ကယား )

P ( at most one tail ) = ?

The set of favourable outcomes = { ( H,H,H,H ),( H,H,H,T ), ( H,H,T,H ),( H,H,T,T )



,( T , T , T , T ) }



16 =

1

4

( 2008 , ႏငငျခား )

P ( getting no tail ) = ?




,( T , T , T , T ) }


P ( getting no tail ) = 1

16


( 1 )P ( exactly one Head ) = ?


, ( H,T,H,H ),( H,T,H,T ),( H,T,T,H ),( H,T,T,T ),( T,H,H,H ),( T,H,H,T ),( T,H,T,H )

,( T,H,T,T ),( T,T,H,H ),( T,T,H,T ),( T,T,T,H ),( T,T,T ,T ) }

( ဖယရမညတနဖးမားအား မမဘာသာျခစထတေပးပါ။ )

Number of favourable outcomes = .


( 2 )P ( exactly two Head ) = ?






P ( exactly two Head ) = 16

( 3 ) P ( exactly one Tail ) = ?





လေသာတနဖးအားျဖညေပးပါ






( 4 ) P ( exactly two Tail ) = ?






P ( exactly two Tail ) = 16

( 5 ) P ( getting no Head ) = ?






P ( getting no Head ) = 16

( 6 ) P ( just one Tail ) = ?






P ( just one Tail ) = 16

( 6 ) P ( just two Tail ) = ?






P ( just two Tail ) = 16

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( 2010 , မေကြး )

A coil is tossed and a die is throw. Head or Tail and a number turn up are recorded

each time. Draw a tree diagram and list possible outcomes. Find the probability that

Head and odd number turn up.

Tossing Co[n 2nd Thrown Possible Outcomes

1 ( H , 1 )

2 ( H , 2 )

3 ( H , 3 )

4 ( H , 4 )

5 ( H , 5 )

6 ( H , 6 )

1 ( T , 1 )






H



2 ( T , 2 )

3 ( T , 3 )

4 ( T , 4 )

5 ( T , 5 )

6 ( T , 6 )

The set of all possible outcome = { ( H , 1 ),( H , 2 ),( H , 3 ),( H , 4 ),( H , 5 )

( H , 6 ),( T , 1 ),( T , 2 ),( T , 3 ),( T , 4 ),( T , 5 ),( T , 6 ) }


P ( Head and odd number turn up ) = ?


( H , 6 ),( T , 1 ),( T , 2 ),( T , 3 ),( T , 4 ),( T , 5 ),( T , 6 ) }


Number of possible outcome =

P ( Head and odd number turn up ) = 12

Ans ; 1

4

( 2010 , ရခင )

P ( Head and even number turn up ) = ?


( H , 6 ),( T , 1 ),( T , 2 ),( T , 3 ),( T , 4 ),( T , 5 ),( T , 6 ) }


Number of possible outcome =

P ( Head and even number turn up ) = 12

Ans ; 1

4

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T



Eg 4. Three tennis players A,B and C play each other only once. The probability that

A will beat B is 1

3 , B will beat C is

2

5 and C will beat A is

2

7 . Calculate the

probability that C win both game.

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