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Chapter 14: Problem 30

1 is 0.8 . If you get contract #1, the probability you also get contract #2 will be 0.鈥� # Your company bids for two contracts. You believe the probability you get contract #1 is 0.8 . If you get contract #1, the probability you also get contract #2 will be 0.2, and if you do not get #1, the probability you get #2 will be 0.3 a. Are the two contracts independent? Explain. b. Find the probability you get both contracts. c. Find the probability you get no contract. d. Let \(X\) be the number of contracts you get. Find the probability model for \(X\) e. Find the expected value and standard deviation.

Short Answer

Expert verified
a. The two contracts are not independent because the probability of getting contract #2 is conditional on whether or not you get contract #1. b. The probability of getting both contracts is 0.16. c. The probability of getting no contracts is 0.14. d. The probability model for X (the number of contracts you get) is: P(X=0) = 0.14, P(X=1) = 0.7, P(X=2) = 0.16. e. The expected value of X (the number of contracts you get) is 1.02 and the standard deviation is 0.44.

Step by step solution

01

Determining Independence

Two events are independent if the occurrence of one doesn't affect the occurrence of the other. Here, the probability of getting contract #2 is dependent on whether contract #1 is obtained or not. Thus, the two contracts are not independent.
02

Probability of Both Contracts

We compute the probability of getting both contracts as follows: P(get both) = P(get #1) * P(get #2|#1). This gives us 0.8 * 0.2 = 0.16.
03

Probability of No Contract

The probability of getting no contract can be computed as follows: P(no contract) = P(not getting #1) * P(no #2|no #1). This gives (1-0.8) * (1-0.3) = 0.14.
04

Probability Model

We can construct the probability model for \(X\) (number of contracts) as follows: P(X=0) = 0.14 , P(X=1) = P(get #1 and no #2) + P(no #1 and get #2)= 0.8 * 0.8 + 0.2 * 0.3 = 0.7 , P(X=2) = 0.16.
05

Expected value and Standard Deviation

First, compute the expected value E(X) as follows: E(X) = 0 * P(X=0) + 1 * P(X=1) + 2 * P(X=2) = 0 + 0.7 + 2*0.16 = 1.02. Then compute the variance Var(X) as follows: Var(X) = (0-E(X))^2 * P(X=0) + (1-E(X))^2 * P(X=1) + (2-E(X))^2 * P(X=2) = 0.196. The standard deviation SD(X) is the square root of the variance, so SD(X) = \(\sqrt{0.196}\) = 0.44.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Independent Events
Understanding the concept of independent events is crucial when analyzing complex probabilistic scenarios. Independent events are those whose occurrences are in no way affected by each other. A prime example would be flipping a coin multiple times; the result of one flip does not influence the result of the next.

However, in the problem given to us, when examining the likelihood of obtaining two contracts, one after the other, we observe that the probability of securing the second contract is affected by the outcome of the first. This interdependence implies that the probability of getting contract #2 shifts depending on whether contract #1 is won (0.2 if #1 is gained) or lost (0.3 if #1 is not gained). Since the outcome of getting contract #1 influences how likely it is to get contract #2, these events are not independent.
Expected Value
The concept of expected value is often synonymous with the 'mean' in probability and serves as a measure to foresee the center of a probability distribution. In simpler terms, it's the average outcome you would anticipate if you could repeat an experiment numerous times.

To calculate the expected value, multiply each potential outcome by its respective probability and then sum all these products together. For instance, if a die is rolled, you'd multiply the value of each side (1 through 6) by the probability of it landing (1/6), and then add these up to find the expected value.

In our problem, we identified the expected number of contracts by using the probabilities of obtaining none, one, or both contracts, weighed by the number of contracts gained in each case. The calculation led us to an expected value of 1.02 contracts, implying that on average, the company can expect to obtain just over one contract.
Standard Deviation
Standard deviation stands as a key statistical measure, which offers insights into the amount of variation or dispersion present in a set of values. It is particularly useful in conveying how spread out a set of outcomes is from the expected value.

To reach the standard deviation, we start by calculating the variance, which is the average of the squared differences from the mean. Once we have this variance, we take its square root to get the standard deviation. This value helps in understanding the level of risk or volatility; a larger standard deviation indicates more spread and thus, higher uncertainty.

In our exercise, after computing the expected value, we calculated the variance by considering the squared differences of each outcome from the expected value, factoring in their probabilities. We found a variance of 0.196, and consequently, a standard deviation of 0.44. This relatively small standard deviation suggests that outcomes will frequently be close to the expected average number of contracts won, signifying lower volatility in the number of contracts won by the company.

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