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# Expected value & Binomial probability

1. Suppose a single die is rolled once. You win \$5 if a 1 or a 3 comes up, win \$12 if a 5 comes up and lose \$10 if a 2, 4, or 6 come up.

a) Complete the table that gives the expected values of each event occurring.

Event x (random variable) P(x) xP(x)
Rolling a 1 +5
Rolling a 2 -10
Rolling a 3 +5
Rolling a 4 -10
Rolling a 5 +12
Rolling a 6 -10

b) Use your table to find the expected value of the game.

c) What does the expected value tell you about the game?

2. Use the below problem to answer the following questions:

A farmer plants 12 saplings. On average, 15% of saplings planted fail to survive (die) their first winter.

a) What type of probability distribution is presented?

b) Find the probability that exactly 5 of his saplings will die in that first winter. Round to 4 decimal places.

c) Find the probability that 5 or less of saplings will die in that first winter. Round to 4 decimal places.

d) Find the probability that more than 5 of saplings will die in that first winter. Round to 4 decimal places.

3. Use the following information to answer the problems below:

Ten percent of American adults are left-handed. Suppose we have a sample of 25 people. Assume that each person in the sample is either left or right-handed.

a) Find the mean and standard deviation for the number of left-handed people in the sample.

b) Would it be unusual to survey a group of 25 people and find that 7 of them are left-handed? Explain.

#### Solution Preview

Please see the attachments.

1. Suppose a single die is rolled once. You win \$5 if a 1 or a 3 comes up, win \$12 if a 5 comes up and lose \$10 if a 2, 4, or 6 come up.

a. Complete the table that gives the expected values of each event occurring.

Event x (random variable) P(x) xP(x)
Rolling a 1 +5 1/6 5/6
Rolling a 2 -10 1/6 -10/6
Rolling a 3 +5 1/6 5/6
Rolling a 4 -10 1/6 -10/6
Rolling a 5 +12 1/6 12/6 = 2
Rolling a 6 -10 1/6 -10/6
Total 1 -8/6

2. Use your table to find the expected value of the game.

Expected value of the game, E (X) =

= -8/6

= -1.33333333

a. What does the expected value tell you about the game?

The expected value tells about the average return from the game. Here since the expected value is negative, there will be no profit and the game results in loss.

2. Use the below problem to answer the following questions:

A farmer plants 12 saplings. On average, 15% of saplings planted fail to survive (die) their first winter.

a) What type of probability distribution is ...

#### Solution Summary

The solution provides step by step method for the calculation of expected value and binomial probabilities. The solution also provides step by step method for the calculation of mean and standard deviation of a binomial distribution. Formula for the calculation and Interpretations of the results are also included.

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