1­Expected values.notebook

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December 15, 2009

Dec 13­8:13 PM

Unit 6 ‐ Probability Distributions

• Probability Distributions• Expected value• Binomial Distribution• Normal Distribution Approximation to

Binomial Distribution• Distribution types

Dec 12­8:41 AM

Probability distribution

Expected value

1­Expected values.notebook

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December 15, 2009

Dec 15­4:42 AM

Probability Distribution

A probability distribution is a table, formula or graph that provides the probabilities of a random variable assuming any of it possible values.

Ex. The probability distribution of the sum of two dice

The sum of the probability of all options is 1.

n(S)

Dec 14­9:05 AM

The expected value is the average value of the random variable over time.

Expected value

like the weighted average!

Ex. The probability distribution of the sum of two dice

In a probability distribution that has n outcomes, {x1, x2,...,xn},the expected value is E(X) = P(X=x1) x1 + P(X=x2) x2 + ... + P(X=xn) xn

E(X)

Note that the expected value of a discrete random variable is not necessarily an integer.

1­Expected values.notebook

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December 15, 2009

Dec 14­9:35 AM

Ex: One trial of an experiment is to toss three coins. What is the expected number of heads?

The expected value of a discrete random variable is not necessarily an integer OR one of the possible outcomes

coin 1 coin 2 coin 3

8 possible permutations

X

0 heads

1 heads

2 heads

3 heads

n(X)

1

3

3

1

P(X)

1/8

3/8

3/8

1/8

E(X) = 1/8 (0) + 3/8 (1) + 3/8 (2) + 1/8 (3) = 3/2

Use a tree diagram to analyse the situation

Group outcomes by the number of heads (X)

Calculate expected value

Dec 15­5:30 AM

Ex: Consider a simple game in which you roll a single die. If you roll an even number, you gain that number of points. If you roll an odd number, you lose that number of points.

a) Show the probability distribution of points for each roll in this game.

b) What is the expected number of points for each roll?

roll points P(X) (X)

123456

‐1+2‐3+4‐5+6

Analyze the situation

‐5 ‐4 ‐3 ‐2 ‐1 0 +1 +2 +3 +4 +5 +6

1/61/61/61/61/61/61/6

Draw probability distribution

1­Expected values.notebook

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December 15, 2009

Dec 15­5:32 AM

a) Show the probability distribution for the number of NDPs on the committee.

c) What is the probability of selecting at least one NDP?

b) What is the expected number of NDPs on the committee?

Ex: An advisory committee has three people. The committee is selected from a groupof 5 NDPs, 6 Liberals and 1 Green Party member.

Analyze the situationX

number of NDPs of committee

n(X) P(X)

S(X)

Dec 13­8:07 PM

HW: pg 277 # 1, 4, 5, 7, 15c, 16b