# Conditional Probability & Independence

Problem #38 review question need help

The key formula on the second page is the formula needed to use to complete number 38. This is supplementary exercise to help me. Solution and explanation.

38. A Morgan Stanley Consumer Research Survey sampled men and women and asked each whether the preferred to drink plain bottled water or a sports drink such as Gatorade or Propel Fitness water (The Atlanta Journal -Constitution, December 28, 2005). Suppose 200 men and 200 women participated in the study, and 280 reported they preferred plain bottled water. Of the group preferring a sports drink, 80 were men and 40 were women.

Let

M = the event the consumer is a man

W = the event the consumer is a woman

B = the event the consumer Preferred plain bottled water

S = the event the consumer Preferred sports drink.

a. What is the probability a person in the study preferred plain bottled water?

b. What is the probability a person in the study preferred a sports drink?

c. What are the conditional probabilities P (M|S) and P (W|S)?

d. What are the joint probabilities P (M∩S) and P (W∩S)?

e. Given a consumer is a man, what is the probability he will prefer a sports drink?

f. Given a consumer is a woman, what is the probability she will prefer a sports drink?

g. Is preference for a sports drink independent of whether the consumer is a man or woman? Explain using probability information.

Please answer A through G with solution and explanation.

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#### Solution Summary

The solution provides step by step method for the calculation of probabilities and conditional probabilities. Formula for the calculation and Interpretations of the results are also included.

Probability: Independence and Conditional Independence

A person tried by a 3-judge panel is declared guilty if at least 2 judges cast votes of guilty. Suppose that when the defendant is in fact guilty, each judge will independently vote guilty with probability 0.7. whereas when the defendant is, in fact, innocent, this probability drops to 0.2. If 70 percent of defendants are guilty, compute the conditional probability that judge number 3 votes guilty given that:

(a) judges 1 and 2 vote guilty:

(b) judges 1 and 2 cast 1 guilty and 1 not guilty vote:

(c) judges 1 and 2 both cast not guilty votes.

Let Ei i = 1, 2, 3 denote the event that judge 1 casts a guilty vote. Are these events independent? Are they conditionally independent? Explain.