Math 155 - Cooley Finite Math with Applications OCC
Section 8.5 – Probability Distributions; Expected Value
Random Variable
A random variable is a function that assigns a real number to each outcome of an experiment.
Probability Distribution
A table that lists the possible values of a random variable, together with the corresponding probabilities.
Expected Value
Suppose the random variable x can take on the n values x1, x2, x3,…, xn. Also, suppose the probabilities that these values occur are, respectively, p1, p2, p3,…, pn. Then the expected value of the random variable is
.
Expected Value for Binomial Probabilty
For binomial probability, . In other words, the expected number of successes is the number of trials times the probability of success in each trial.
J Exercises:
1) Roulette is a common game of gambling found in Casinos. An American roulette wheel contains 38
numbers: 18 are red, 18 are black, and 2 are green. When the roulette wheel is spun, the ball is equally
likely to land on any of the 38 numbers. Suppose that you bet $1 on red. If the ball lands on a red
number, you win $1; otherwise you lose your $1. Let x be the amount you win on your $1 bet. Then x
is a random variable whose probability distribution is as follows:
x / 1 / –1P(x) / 0.474 / 0.526
a) Verify that the probability distribution is correct.
b) Find the expected value of the random variable x.
c) On average, how much will you lose per play?
J Exercises:
2) The following is a histogram with the random variable x representing the number of tails showing when
three fair coins are tossed:
What is the expected value for the number of tails showing? (Hint: There are two ways you can do this.)
3) In a club with 20 senior and 10 junior members, what is the expected number of junior members on a 4-
member committee?
J Exercises:
4) Suppose 5 apples in a barrel of 25 apples are known to be rotten.
a) Draw a histogram for the number of rotten apples in a sample of 2 apples.
b) What is the expected number of rotten apples in a sample of 2 apples?
J Exercises:
5) A raffle offers a first prize of $1000, 2 second prizes of $300 each, and 20 third prizes of $10 each. If
10,000 tickets are sold at 50ȼ each, find the expected payback for a person buying one ticket. Is this a fair game?
6) In one form of the game Keno, the house has a pot containing 80 balls, each marked with a different
number from 1 to 80. You buy a ticket for $1 and mark one of the 80 numbers on it. The house then selects 20 numbers at random. If your number is among the 20, you get $3.20 (for a net winning of $2.20). Find the expected payback for playing one game of Keno.
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