Question: In a survey, some of the questions concern sensitive issues (e.g., income, drug use, sexual experiences).  As a result, some respondents do not answer the questions truthfully.  Suppose that 30% of the members of a particular population had incomes over $100,000 last year.  A random sample of n = 10 members of this population is taken, and each person in the sample is asked "Was your income over $100,000 last year?"  If a person really had an income over $100,000, the probability that he or she will give a truthful answer to this question is 0.7.  If a person’s income was not over $100,000, the probability that he or she will give a truthful answer is 0.9.

1. Find the probability distribution of X , the number of people in the sample who had incomes over $100,000 last year.
2. Find the probability distribution of Y , the number of "yes" answers to the question in the sample of size 10.
3. Find the probability distribution of T , the number of truthful answers from the 10 respondents.
4. Given the answer "Yes, my income was over $100,000," what is the probability that the income of this person was, indeed, over $100,000? Similarly, given the answer "No, my income was not over $100,000," what is the probability that the income of this person was, indeed, below $100,000?

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