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

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?


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 =


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}


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


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


‘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


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


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


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


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


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



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



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


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


The Product Rule

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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