Chapter 5, Lesson 3, Developing Discrete Probability Distributions Theoretically & Finding Expected Values

The Expected Value

is the sum of (X times P(X)) for all values of a random variable X, or E(X) = The sum of X * P(X)

probability distribution

summarizes the probability for a particular experiment.

A roulette wheel has 38 slots labeled with the numbers 1 through 36 and then 0 and 00. Slots 1 through 36 are colored either red or black. There are 18 red and 18 black. Slots 0 and 00 are colored green (see picture). Suppose that for a bet of $1 on red, the casino will pay you $2 if the ball lands on a red slot (a net gain of $1), and otherwise you lose your dollar. What can you expect to win or lose in this game?

I expect to lose about 5 cents per game. (The probability of the ball landing on red is 18/38. This is worth +$1 to you ($2 winnings - $1 bet). The probability of the ball not landing on red is 1 - 18/38 = 20/38. This is worth $-1 to you (you lose the dollar you bet). Using the expected value formula we get: (18/38 x 1) + (20/38 x -1) = -0.05263, or a loss of about a 5 cents per game.)

Suppose you play a game where you spin a spinner (see picture below) with areas of the colors on the spinner broken down as shown: 10% blue, 60% green, and 30% red. In addition, if the spinner lands on red you win 6 points, if it lands on blue you win 1 point, and if it lands on green you lose 5 points. If you keep spinning, how many points can you expect to win or lose per spin?

I expect to lose 1.1 points per spin.

You play a game where you toss a coin. On each toss if it lands with heads up, you win $1. However, if it lands with tails up, you lose $2. If you continue to play this game, how much can you expect to win or lose per game?

I expect to lose 50 cents per game. (The probability of heads is 1/2 and this is worth +$1 to you. The probability of tails is 1/2 and this is worth -$2 to you. So we multiply the value times the probability and then add: (1/2 x 1) + (1/2 x -2) = 0.5 - 1 = -0.5. So you expect to lose $0.50 per game.)

The Expected Value This is the expected value of X. In other words, we expect one head when we toss two coins.

E(X) = (0 * 0.25) + (1 * 0.5) + (2 * 0.25)

Suppose you toss three coins. What is the probability that you get two heads and one tail if the order in which you get them does not matter?

3/8 (Any time you toss a coin, the probability you get one side is 1/2. Thus, for any outcome in tossing three coins the probability is 1/2 x 1/2 x 1/2 = 1/8. There are three ways to get get two heads (H) and a tail (T) if you are not interested in the order in which it happens: HHT, HTH and THH. For each outcome the probability is 1/8. Since there are three ways that this can happen, the probability of two heads and a tail is 3 x 1/8 = 3/8.)

A roulette wheel has 38 slots labeled with the numbers 1 through 36 and then 0 and 00. Slots 1 through 36 are colored either red or black. There are 18 red and 18 black. Slots 0 and 00 are colored green (see picture). On one spin of the roulette wheel, what is the probability that the ball lands on a red slot?

9/19 (Since there are 38 slots, and 18 of them are red, the probability of the ball landing on a red slot is 18/38 which simplifies to 9/19.)