Question

A contractor has submitted bids on three state jobs: an office building, a theater, and a parking garage. State rules do not allow a contractor to be offered more than one of these jobs. If this contractor is awarded any of these jobs, the profits earned from these contracts are: 13 million from the office building, 9 million from the theater, and 4 million from the parking garage. His profit is zero if he gets no contract. The contractor estimates that the probabilities of getting the office building contract, the theater contract, the parking garage contract, or nothing are .17, .27, .45, and .11, respectively. Let x be the random variable that represents the contractor's profits in millions of dollars. Write the probability distribution of x. Find the mean and standard deviation of x.

Answer 1

Probability distribution:

x (million) P(x)

13 0.17

9 0.27

4 0.45

0 0.11

Mean: 6.44

Standard deviation: 4.04

Step-by-step explanation:

The probability distribution is a table that shows the profits earned and its respective probabilities, so:

x (million) P(x)

13 0.17

9 0.27

4 0.45

0 0.11

Then, the mean can be calculated as the sum of each profit multiplied by its respective probability. Therefore, the mean E(x) is equal to:

E(x) = 13(0.17) + 9(0.27) + 4(0.45) + 0(0.11)

E(x) = 2.21 + 2.43 + 1.8 + 0

E(x) = 6.44

Finally, to calculate the standard deviation, we first need to find the differences between each value and the mean, and then find the square of these values, so:

x x - E(x) (x - E(x))²

13 13 - 6.44 = 6.56 (6.56)² = 43.03

9 9 - 6.44 = 2.56 (2.56)² = 6.55

4 4 - 6.44 = -2.44 (-2.44)² = 5.95

0 0 - 6.44 = -6.44 (-6.44)² = 41.47

Then, the standard deviation will be the square root of the sum of the values in the last column multiply by each probability:

`\begin{gathered} s=\sqrt[]{43.03(0.17)+6.55(0.27)+5.95(0.45)+41.47(0.11)} \\ s=\sqrt[]{16.3264} \\ s=4.04 \end{gathered}`

Therefore, the answers are:

Probability distribution:

x (million) P(x)

13 0.17

9 0.27

4 0.45

0 0.11

Mean: 6.44

Standard deviation: 4.04