Aim: Or Probabilities Course: Math Lit.
Do Now:
Aim: What are ‘Or’ Probabilities?
What is the probability of spinning a number greater than 8 or an odd number?
Aim: Or Probabilities Course: Math Lit.
Probability of A or B
What is the probability of spinning a number greater than 8 or an odd number?
1211
10
9
8
7 6
5
4
3
21 Count the number of
successes for
n > 8
n - odd not yet counted
9, 10, 11, 12 4
1, 3, 5, 7 4
(successes)(greater than 8 or odd)
( )
nP
n S
8
12
{1, 3, 5, 7, 9, 10, 11, 12}> 8 odd =
Aim: Or Probabilities Course: Math Lit.
Union
Region I
Region II
Region IV
A B
Reg. III
U
The union of sets A and B is region II, II & IV.
‘or’ is the term used to describe union
The union of sets A and B, denoted by A B, is the set consisting of all elements of A or B or both.
A B = {x|x A OR x B}
Aim: Or Probabilities Course: Math Lit.
Union of Do Now
> 8 = {9, 10, 11, 12}
U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
odd = {1, 3, 5, 7, 9, 11}
> 8
9 11
odd
U
1, 3, 5, 7, 10, 12
2, 4, 6, 8
A B = {1, 3, 5, 7, 9, 10, 11, 12}
n(A B) = 8( ) 8
(greater than 8 or odd)( ) 12
n A BP
n U
Aim: Or Probabilities Course: Math Lit.
Independent Events
Mutually exclusive – two events A & B are mutually exclusive if they can not occur at the same time. That is, A and B are mutually exclusive when A B =
An outcome for A or B is in one or the other. If the events are mutually exclusive the
P(A or B) = P(A) + P(B)
If one card is randomly selected from a deck of cards, what is the probability of selecting a king or a queen? mutually
exclusive?yes
( ) ( ) ( )
4 4 8 2
52 52 52 13
P king or queen P k P q
Aim: Or Probabilities Course: Math Lit.
‘Or’ Probabilities Not Mutually Exclusive
From a standard deck you randomly select one card. What is the probability of selecting a diamond or a face card? mutually
exclusive?no
13( )
52P diamonds
12( )
52P facecards
common elements A B
n(A B) = 3 {K, Q, J}
P(or fcd) = 13 12 25
52 52 52
13 12 3 22
52 52 52 52
Aim: Or Probabilities Course: Math Lit.
Probability of (A or B)
Example: Find the probability of rolling a die and getting a number that is odd or greater than 2.
{1,3,5} {3,4,5,6}
( ) 3(odd)
( ) 6
n EP
n S
( ) 4( 2)
( ) 6
n EP
n S
successes
(odd) ( 2) (odd >2)(odd >2)
( ) ( ) ( )
n n nP
n S n S n S
3 4 2 5(odd >2)
6 6 6 6P
P(A or B) = P(A) + P(B) - P(A and B)
P(A B) = P(A) + P(B) - P(A B)
P(A B) = n(A) + n(B) - n(A B) n(S) n(S) n(S)
If A and B are not mutually exclusive events, then
Aim: Or Probabilities Course: Math Lit.
Model Problem
Find the probability of rolling a die and getting a number that is odd or greater than 2.
24
6
1 3
5
4
6
3
5
1 3
5
odd > 2
P(odd) = 3/6 P(> 2) = 4/6
A B = {1, 3, 4, 5, 6}
n(A B) = 5
( ) 5(> 2 or odd)
( ) 6
n A BP
n U
n(U) = 6
Aim: Or Probabilities Course: Math Lit.
Model Problem
In a group of 50 students, 23 take math, 11 take psychology, and 7 take both. If one student is
selected at random, find the probability that the student takes math or psychology
23( )
50P M
11( )
50P Psy
7( )
50P M Psy
P(A B) = P(A) + P(B) - P(A B)
23 11 7 27( )
50 50 50 50P M Psy
23416 7
M Psy
Aim: Or Probabilities Course: Math Lit.
Model Problems
1. A card is drawn from a standard deck of 52. Find P(Ace or jack)
452P(ace) = 4
52P(jack) =
P(ace or jack) = 452
452
+ 852
=
2. A card is drawn from a standard deck of 52. Find P(king or face card)
452P(king) = 12
52P(face) =
P(king or face) = 452
1252
+ 452
_ 1252
=
mutually exclusive
not mutually exclusive
Aim: Or Probabilities Course: Math Lit.
Model Problems
In drawing a card from the deck at random, find the probability that the card is:
A. A red king
B. A 10 or an ace
C. A jack or a club
A red king must be red and a king
P(red and king) = 252
There are 4 jacks and 13 clubs, but one of the cards is both (jack of clubs)
P(jacks or clubs) = 452
1352
+ 152
_ 1652=
10’s and aces have no common outcomes
P(10’s or aces) = 452
452
+ 052
_ = 852
P(A and B) = P(A) · P(B)
26 4 1
52 52 26
P(A B) = P(A) + P(B) - P(A B)
P(A B) = P(A) + P(B) - P(A B)
mutually exclusive
not mutually exclusive
Aim: Or Probabilities Course: Math Lit.
Model Problems
Based on the table below, if one person is randomly selected from the US military, find the probability that this person is in the Army or is a woman.
Air Force
Army Marines Navy Total
Male 290 400 160 320 1170
Female 70 70 10 50 200
Total 360 470 170 370 1370
Active Duty US Military Personnel, in 000’s
P(A B) = P(A) + P(B) - P(A B)
P(Army Female) = P(A) + P(F) - P(A F)470
1370
200
1370
70
1370
60
137
not mutually exclusive
Aim: Or Probabilities Course: Math Lit.
1. The probability of an impossible event is 0.
2. The probability of an event that is certain to occur is 1.
3. The probability of an event E must be greater than or equal to 0 and less that or equal to 1.
4. P(A and B) = n(A B) n(S)
5. P(A or B) = P(A) + P(B) - P(A B)
6. P(Not A) = 1 - P(A)
7. The probability of any even is equal to the sum of the probabilities of the singleton outcomes in the event.
8. The sum of the probabilities of all possible singleton outcomes for any sample space must always equal 1.
Probability Rules
Aim: Or Probabilities Course: Math Lit.
Model Problems
Five more men than women are riding a bus as passengers. The probability that a man will be the first passenger to leave the bus is 2/3. How many passengers on the bus are men, and how many are women?
x = number of womenx + 5 = number of men
2x + 5 = number of passengers
P(man) = Number of menNumber of passengers
x + 52x + 5
23
= 4x + 10 = 3x + 15x = 5
x + 5 = 10
510
Aim: Or Probabilities Course: Math Lit.
The Product Rule