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Recap: Independent Events

Example 1: I travel by bus and train on my journey to work. Given that P(bus late) is .! and

independently P(train late) is ."# $al$ulate the probability that:a) both the bus and the train are late%

b) either the bus or the train (or both) are late.

&e $an answer this 'uestion by drawin a tree diaram. et * + bus late# , + train late.

a) P(* and ,) + .! - ." + .

b) P(* /0 ,) + 1 P(neither are late) + 1 .2 - .3 + .44.

Example ": 5aroline plays a ame with a 6air $oin. 7he keeps throwin the $oin until a head is

obtained# at whi$h point she stops. 5al$ulate the probability that:

a) the ame 6inishes on the !rdthrow%

b) the ame 6inishes in less than 8 attempts%

$) the ame $ontinues 6or more than 1 throws.

a) P(ame 6inishes on !rd

o) + P(,ail ,9E ,ail ,9E 9ead) +

1 1 1 1

" " " 3 =b) P(ame 6inishes in less than 8 attempts) + P(6inishes on 1stor "nd or !rd or 4th o)

*ut# P(ame 6inishes on 1sto) + P(9ead) +1

"

P(ame 6inishes on "ndo) + P(, ;< 9) +1 1 1

" " 4 =

P(ame 6inishes on 4th o) + P(,# ,# ,# 9) +1 1 1 1 1

" " " " 1 =

7o P(ame 6inishes in less than 8 oes) +1 1 1 1 18

" 4 3 1 1

+ + + =

$) P(ame $ontinues 6or more than 1 throws) + P(6irst 1 throws are tails) +

1

1 1

" 1"4

=

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Dependent Events and Conditional Probability

Introdu$tory example: ,he probability that it will be sunny tomorrow is1

!. I6 it is sunny# the

probability that 7usan plays tennis is4

8. I6 it is not sunny# the probability that 7usan plays tennis is

"

8. =ind the probability that 7usan plays tennis.

P(tennis) + P(sunny ;< tennis) > P(not sunny ;< tennis)

+1 4 " " 3

! 8 ! 8 18 + =

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Example ": /n averae I et up early ! days out o6 1# and et up late one day in 1.

I 6oret somethin on " out o6 every 8 days on whi$h I am late and on one third o6 the days on whi$h

I am early. I do not 6oret anythin on the days when I et up on time.

(a) =ind the probability that I 6oret somethin.

(b) 5on6irm that 6or 3? o6 the time I am neither 6oret6ul nor late.

et E + et up early, + et up on time

+ et up late

= + 6oret somethin

=@ + don@t 6oret anythin

,he tree diaram then looks as 6ollows:

a) P(6oret somethin) + P(E and =) > P(, and =) > P( and =)

+! 1 1 " 2

1 ! 1 1 8 8

+ + =

b) ,he two situations where I am neither 6oret6ul nor late have been indi$ated on the tree diaram:7o P(neither 6oret6ul o6 late) +

! " 41

1 ! 1 8 + =

,here6ore# on 3? o6 o$$asions I will neither 6oret anythin nor be late.

A part (b)

A part (b)