Lesson9.3

Probability

Precalculus

Quick Review

How many outcomes are possible for the following experiments.

1. Two coins are tossed.

2. Two different 6-sided dice are rolled.

3. Two chips are drawn simultaneously without replacement from

a jar wi

4 2

8 2

th 8 chips.

4. Two different cards are drawn from a standard deck of 52.

5. Evaluate without using a calculator. C

C

436

8 2C

13264!

2!2!8!

2!6!

6

28

3

14

28

What you’ll learn about

Sample Spaces and Probability FunctionsDetermining ProbabilitiesVenn Diagrams and Tree DiagramsConditional ProbabilityBinomial Distributions

… and whyEveryone should know how mathematical the “laws of chance” really are.

Probability of an Event (Equally Likely Outcomes)

If is an event in a finite, nonempty sample space

of equally likely outcomes,

then the of the event

is

E S

Eprobability

the number of outcomes in ( ) .

the number of outcomes in

EP E

S

Probability Distribution for the Sum of Two Fair Dice

Outcome Probability2 1/363 2/364 3/365 4/366 5/367 6/368 5/369 4/3610 3/3611 2/3612 1/36

Example Rolling the Dice

Find the probability of rolling a sum divisible by 4 on a singleroll of two fair dice.

To get the probability of

we add up the probabilities of the outcomes in :

E

E

The event consists of the outcomes 4,8,12 . E

3 5 1( )

36 36 36P E

9

36

1

4

Probability Function

A is a function that assigns

a real number to each outcome in a sample space

subject to the following conditions:

1. 0 ( ) 1;

2. the sum of the probabilities of all outcomes in

P

S

P O

probability function

is 1;

3. ( ) 0.

S

P

Probability of an Event (Outcomes not Equally Likely)

Let be a finite, nonempty sample space in which

every outcome has a probability assigned to it by a

probability function . If is any event in ,

the of the event is the sum of the

p

S

P E S

Eprobability

robabilities of all the outcomes contained in . E

Strategy for Determining Probabilities

3. If the sample space does not have equally likely outcomes,

determine the probability function. (This is not always easy to do.)

Check to be sure that the conditions of a probability function are satisfied.

Then the probability of an event is determined by adding up the probabilities

of all the outcomes contained in .

E

E

1. Determine the sample space of all possible outcomes.

When possible, choose outcomes that are equally likely.

2. If the sample space has equally likely outcomes,

the probability of an event is determined by

the number of outcomes in counting: ( ) .

the number of outcomes in

E

EP E

S

Example Choosing Chocolates

Dylan opens a box of a dozen chocolate cremes and offers three of them to Russell. Russell likes vanilla cremes the best, but all the chocolates look alike on the outside. If five of the twelve cremes are vanilla, what is the probability that all of Russell’s picks are vanilla?

5 3There are 10 ways.C

The experiment in question is the selection of three chocolates,

without regard to order, from a box of 12.

12 3There are 220 outcomes of this experiment.C

The event consists of all possible combinations of 3 that can be chosen,

without regard to order, from the 5 vanilla cremes available.

E

10Therefore, ( )

220P E

1

22

Multiplication Principle of Probability

Suppose an event A has probability p1 and an event B has probability p2 under the assumption that A occurs. Then the probability that both A and B occur is p1p2.

Example Choosing ChocolatesDylan opens a box of a dozen chocolate cremes and offers three of

them to Russell. Russell likes vanilla cremes the best, but all the chocolates look alike on the outside. If five of the twelve cremes are vanilla, what is the probability that all of Russell’s picks are vanilla?

60a vanilla creme on all three pi

5 4 3 1

12 11 1cks is

13200 22.

The probability of picking a vanilla creme on the first draw is 5/12.

Under the assumption that a vanilla creme was selected in the first

draw, the probability of picking a vanilla creme on the second draw is

4/11. Under the assumption that a vanilla creme was selected in the first

and second draw, the probability of picking a vanilla creme on the third

draw is 3/10. By the Multiplication Principle, the probability of picking

Conditional Probability Formula

If the event depends on the event ,

( and )then ( | ) .

( )

B A

P A BP B A

P A

Binomial Distribution

1

1 1

n

n

n p

nn p q

n

1

1

0

r n

n

np q

r

q

Suppose an experiment consists of -independent repetitions of an

experiment with two outcomes, called "success" and "failure." Let

(success) and (failure) . (Note that 1 .)

Then the terms in th

n

P p P q q p e binomial expansion of ( ) give the respective

probabilities of exactly , 1,..., 2, 1, 0 successes.

np q

n n

Number of successes out of Probability independent repetitionsn

Example Shooting Free Throws

Suppose Tommy makes 92% of his free throws. If he shoots 15 free throws, and if his chance of making each one is independent of the other shots, what is the probability that he makes all 15?

(15 successes)P 0.286 15

0.92

Example Shooting Free Throws

Suppose Tommy makes 92% of his free throws. If he shoots 15 free throws, and if his chance of making each one is independent of the other shots, what is the probability that he makes exactly 10?

10 515= 0.92 0.08

10

(10 successes)P 0.00427

Homework:

Text pg728/729 Exercises

# 2-8 even, 12-24 (intervals of 4)