Mean and Standard Deviation for Discrete Probability Distributions

1. A nationwide coffee chain is downsizing its branches in six states. It has already told the store managers in five of the six states the amount of downsizing that will occur. The store managers in the sixth state know that the downsizing that has occurred in the other states has the following distribution. If they assume that the probabilities in this distribution represent the likelihood that their stores will be downsized by the given percentages, what percentage of their employees could they expect to be downsized, and with how much variation?

Coffee Chain Downsizing

Percentage Downsized

Probability

10%

0.25

15%

0.35

20%

0.25

25%

0.1

40%

0.05

1. Suppose that a friend is helping put on a fundraiser for the local animal shelter. One activity is a game using a bowl that contains six green marbles and eight blue marbles. To play the game, each person draws two marbles without replacement (and without looking). If both marbles are green, the player wins $25. If not, the player must donate $10 to the animal shelter. The marbles are then replaced for the next player. 
2. Calculate the expected value of the game for the player.
3. Suppose that the marble game is played 350 times. How much money would you expect to be donated to the animal shelter?
4. Begin by creating a probability distribution for the number of heads in five tosses of a coin.
5. Explain why this distribution meets the conditions for a binomial distribution.
6. Use the discrete probability distribution to calculate the expected value of the random variable.
7. The expected value of the random variable is equal to the mean of the distribution. As mentioned in the list of properties of the binomial distribution, the mean of a binomial distribution is given by m = np. Use this formula to calculate the mean of this probability distribution.
8. Does the value you calculated in part b. match the value you calculated in part c.? Explain why or why not.