the laws of probability. basic laws to review probability of an event : probability of an event :...

The Laws of Probability The Laws of Probability

Basic Laws to reviewBasic Laws to review

Probability of an event : Probability of an event :

Complement of an event: P(A’) = 1 – Complement of an event: P(A’) = 1 – P(A)P(A)

The complement of event A is that the The complement of event A is that the event A does not occur.event A does not occur.

space) (sample outcomes possible ofnumber

occurcan event the waysofnumber

If the probability of rolling a If the probability of rolling a ‘2’ on a six sided die is one ‘2’ on a six sided die is one sixth, what’s the probability sixth, what’s the probability of not rolling a 2? (Enter a of not rolling a 2? (Enter a

fraction)fraction)

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Disjoint Events Disjoint Events Two events A and B are disjoint or Two events A and B are disjoint or

mutually exclusive if they have no mutually exclusive if they have no outcomes in common and can never outcomes in common and can never happen simultaneously. Using a Venn happen simultaneously. Using a Venn Diagram disjoint or mutually exclusive Diagram disjoint or mutually exclusive events are shown: events are shown:

A B

In Algebra:

P(A or B)=P(A U B) = P(A) + P(B)

P(A and B) = P(A ∩ B) = 0

Example: Example:

A cooler contains 20 bottles made up A cooler contains 20 bottles made up of 8 cokes, 5 pepsi’s, and 7 waters. of 8 cokes, 5 pepsi’s, and 7 waters. The probability of choosing a coke or The probability of choosing a coke or pepsi is: pepsi is:

20

13

20

5

20

8

You Do: You Do:

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For the junior class picnic, parents prepare hotdogs, hamburgers, or bar – b – que sandwiches. They have 100 hot dogs, 55 hamburgers, and 90 Bar – b – que sandwiches. What is the probability that you will get a hamburger or a bar – b – que sandwich? (Click in your answer as a reduced fraction)

General Addition Rule: General Addition Rule: If two (or more) events are not disjoint If two (or more) events are not disjoint

the above formula doesn’t work. The the above formula doesn’t work. The rule is: P(A or B) = P(A) + P(B) – P(A and rule is: P(A or B) = P(A) + P(B) – P(A and B) B)

EXAMPLE: How many people have a EXAMPLE: How many people have a dog? How many people have a cat? How dog? How many people have a cat? How many people have a dog and a cat? many people have a dog and a cat?

D C

ExampleExample Example: In a class there are 12 boys Example: In a class there are 12 boys

made up of 8 senior and 4 juniors. There made up of 8 senior and 4 juniors. There are also 8 girls, made up of 3 seniors and 5 are also 8 girls, made up of 3 seniors and 5 juniors. Find the probability of choosing a juniors. Find the probability of choosing a boy or a senior. boy or a senior. Note – choosing a boy and choosing a senior are Note – choosing a boy and choosing a senior are

not disjoint (they can occur simultaneously). So not disjoint (they can occur simultaneously). So the probability of choosing a boy or a senior = the probability of choosing a boy or a senior = P(boy) + P(senior) – P(senior boy). P(boy) + P(senior) – P(senior boy).

Boy Boy GirlGirl TotalTotal

SeniorSenior 88 33 1111

JuniorJunior 44 55 99

TotalTotal 1212 88

You Do: You Do:

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In a class there are 12 boys made up of 8 In a class there are 12 boys made up of 8 senior and 4 juniors. There are also 8 senior and 4 juniors. There are also 8 girls, made up of 3 seniors and 5 juniors. girls, made up of 3 seniors and 5 juniors. Find the probability of choosing a girl or a Find the probability of choosing a girl or a junior. (Click in a reduced fraction.)junior. (Click in a reduced fraction.)

Boy Boy GirlGirl TotalTotal

SeniorSenior 88 33 1111

JuniorJunior 44 55 99

TotalTotal 1212 88

A rental car lot has 49 A rental car lot has 49 American made cars and 26 American made cars and 26

Foreign cars. Of the American Foreign cars. Of the American cars, 35 of them are white and cars, 35 of them are white and

of the foreign cars, 15 are of the foreign cars, 15 are white. A car is chosen at white. A car is chosen at

random. Find the probability random. Find the probability that the car is American or that the car is American or

White.White. 64/7564/75

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A rental car lot has 49 A rental car lot has 49 American made cars and 26 American made cars and 26

Foreign cars. Of the American Foreign cars. Of the American cars, 35 of them are white and cars, 35 of them are white and

of the foreign cars, 15 are of the foreign cars, 15 are white. A car is chosen at white. A car is chosen at

random. Find the probability random. Find the probability that the car is Foreign or not that the car is Foreign or not White. (May need a chart) White. (May need a chart)

A pizza shop has two sizes of pizzas, A pizza shop has two sizes of pizzas, large and small. On a certain day, a large and small. On a certain day, a

pizza shop made 59 plain pizza and 72 pizza shop made 59 plain pizza and 72 pizzas with toppings. Of the 59 plain pizzas with toppings. Of the 59 plain pizzas, 19 were small and of the 72 pizzas, 19 were small and of the 72

pizzas with toppings, 42 were large. A pizzas with toppings, 42 were large. A pizza is chosen at random. Find the pizza is chosen at random. Find the probability that the pizza is small or probability that the pizza is small or

plain. (Make a chart)plain. (Make a chart) 89/1389/1311

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A pizza shop has two sizes of pizzas, A pizza shop has two sizes of pizzas, large and small. On a certain day, a large and small. On a certain day, a

pizza shop made 59 plain pizza and 72 pizza shop made 59 plain pizza and 72 pizzas with toppings. Of the 59 plain pizzas with toppings. Of the 59 plain pizzas, 19 were small and of the 72 pizzas, 19 were small and of the 72

pizzas with toppings, 42 were large. A pizzas with toppings, 42 were large. A pizza is chosen at random. Find the pizza is chosen at random. Find the probability that the pizza is large or probability that the pizza is large or

with toppings. (Make a chart)with toppings. (Make a chart)

Independent EventsIndependent Events

Two events are independent if knowing Two events are independent if knowing one occurs doesn’t change the probability one occurs doesn’t change the probability of the other occurring. This is the non – of the other occurring. This is the non – mathematical definition.mathematical definition.

EXAMPLE: Tossing one coin and then EXAMPLE: Tossing one coin and then another coin are independent events. another coin are independent events. The result of the second coin toss has The result of the second coin toss has nothing to do with the results of the first nothing to do with the results of the first coin toss.coin toss.

Proving IndependenceProving Independence

Proving independence is difficult. For Proving independence is difficult. For instance, is making a second foul shot in instance, is making a second foul shot in basketball independent of the first foul basketball independent of the first foul shot? What do you think? shot? What do you think?

Mathematical proof of independence: If Mathematical proof of independence: If two events are independent then:two events are independent then:

P(A) P(B) = P(A and B)P(A) P(B) = P(A and B) ALSO, IF P(A) P(B) = P(A and B), THEN ALSO, IF P(A) P(B) = P(A and B), THEN

the two events are independent. the two events are independent.

ExampleExample Using the previous example, are Using the previous example, are

choosing a senior and choosing a boy choosing a senior and choosing a boy independent?independent?

If so, P(senior) X P(Boy) = P(Senior boy)If so, P(senior) X P(Boy) = P(Senior boy) Fill in the probabilities on the left side Fill in the probabilities on the left side

of equation; of equation; Fill in the probabilities on the right side Fill in the probabilities on the right side

of equation; of equation; Are the two sides equal? Are the two sides equal? What does this tell you? What does this tell you?

At Parkland High School, there are 13 At Parkland High School, there are 13 math teachers. There are 5 men and 9 math teachers. There are 5 men and 9

women. Of the 5 men, 2 have their women. Of the 5 men, 2 have their national board certification and of the national board certification and of the

9 women 1 has her national board 9 women 1 has her national board certification. Is choosing a teacher certification. Is choosing a teacher

with national board certification and a with national board certification and a

woman independent? woman independent? A.A. YesYes

B.B. NoNo

Men Men WomeWomenn

Nat’l BoardsNat’l Boards 22 11 33No Nat’l No Nat’l BoardBoard 33 77 1010

55 88 1313

Conditional ProbabilityConditional Probability

Definition: The probability of one Definition: The probability of one event happening, event happening, given thatgiven that another event has happened. another event has happened.

Notation: P(B|A) means the Notation: P(B|A) means the probability of B occurring given probability of B occurring given that A occurred. that A occurred.

Example Example Boy Boy GirlGirl TotalTotal

SeniorSenior 88 33 1111

JuniorJunior 44 55 99

TotalTotal 1212 88

Find P(Boy|Senior) – the probability of choosing a Boy given that we chose a senior.

There are 11 seniors and 8 of them are boys, so P(B|S) =

Find P(Senior|Boy) – the probability of choosing a senior, given that we chose a boy.

There are 12 boys, and of those 12, 8 are seniors, so P(S|B) =

11

8

12

8

THE FORMULATHE FORMULA

P(Boy)

Boy)P(Senior Boy)|P(Senior

P(Senior)

Boy)P(Senior Senior)|P(Boy

P(A)

B)P(A A)|P(B

12

8

2012208

Boy)|P(Senior

11

8

2011208

Senior)|P(Boy

YOU DO: YOU DO:

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Find the P(Junior|Girl)

Boy Boy GirlGirl TotalTotal

SeniorSenior 88 33 1111

JuniorJunior 44 55 99

TotalTotal 1212 88

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A. Find the P(Junior|Boy)

Boy Boy GirlGirl TotalTotal

SeniorSenior 88 33 1111

JuniorJunior 44 55 99

TotalTotal 1212 88

You Do:

You Do: You Do:

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Find the P(Girl|Senior)

Boy Boy GirlGirl TotalTotal

SeniorSenior 88 33 1111

JuniorJunior 44 55 99

TotalTotal 1212 88

Example: Example: Let a pair of fair dice be tossed. Make a Let a pair of fair dice be tossed. Make a

sample space of each possible toss. sample space of each possible toss.

66 6,16,1 6,26,2 6,36,3 6,46,4 6,56,5 6,66,6

55 5,15,1 5,25,2 5,35,3 5,45,4 5,55,5 5,65,6

44 4,14,1 4,24,2 4,34,3 4,44,4 4,54,5 4,64,6

33 3,13,1 3,23,2 3,33,3 3,43,4 3,53,5 3,63,6

22 2,12,1 2,22,2 2,32,3 2,42,4 2,52,5 2,62,6

11 1,11,1 1,21,2 1,31,3 1,41,4 1,51,5 1,61,6

11 22 33 44 55 66

Example cont…Example cont… Find the probability that at least one of Find the probability that at least one of

the die is a four.the die is a four. Find the probability that the sum is a 7Find the probability that the sum is a 7 Find P(the one of the die is a 4|sum is 7)Find P(the one of the die is a 4|sum is 7) Given that at least one of the dice is 4, Given that at least one of the dice is 4,

find the probability that the sum is 7. find the probability that the sum is 7. Are rolling two dice with at least one of Are rolling two dice with at least one of

them a 4 and the sum being 7 them a 4 and the sum being 7

A) mutually exclusive or B) A) mutually exclusive or B) independent. Why? independent. Why?

A cooler has 12 Coke’s and 15 A cooler has 12 Coke’s and 15 Pepsi’s. 9 of the Cokes are Pepsi’s. 9 of the Cokes are

diet Coke’s and 5 of the Pepsi’s diet Coke’s and 5 of the Pepsi’s are diet Pepsi’s. Make a chart are diet Pepsi’s. Make a chart of what is in the cooler and of what is in the cooler and find the probability that the find the probability that the

bottle is a Coke, given that the bottle is a Coke, given that the bottle is a diet drink. bottle is a diet drink.

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A cooler has 12 Coke’s and 15 A cooler has 12 Coke’s and 15 Pepsi’s. 9 of the Cokes are Pepsi’s. 9 of the Cokes are

diet Coke’s and 5 of the Pepsi’s diet Coke’s and 5 of the Pepsi’s are diet Pepsi’s. Given that the are diet Pepsi’s. Given that the

bottle is a Pepsi, find the bottle is a Pepsi, find the probability that the bottle is a probability that the bottle is a

Diet PepsiDiet Pepsi1/31/3

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Are choosing a regular (non Are choosing a regular (non diet) and choosing a diet drink diet) and choosing a diet drink

independent or mutually independent or mutually exclusive? Why?exclusive? Why?

A.A. IndependentIndependent

B.B. Mutually exclusiveMutually exclusive