MATH 367, Fall 98, CHAPTER 3 Section 1 INDEPENDENCE

1. If A and B are independent, then so are A and \overline{B}.

2. Two fair dice are thrown. Let E denote the event that the sum of the dice is 7. Let F denote the event that the first die equals 4 and let G be the event that the second die equals 3. Study if these events are independent in pairs and/or as a triplet.

3. Four persons enter an elevator. There are 5 floors at which they can get off. The floors at which the passengers get off are independent. Each person is as likely to get off at any one floor as at any other floor. What is the probability that all passengers get off at the same floor?

4. A coin is tossed 5 times. What is the probability of getting two heads?

5. A die is thrown 5 times. What is the probability of:

(a) getting 2 sixes?

(b) getting at least 2 sixes?

(c) it falls six on the 3rd throw?

6. A die is thrown repeatedly till it falls “six”. What is the probability it takes 6 throws? at least six throws?

7. A die is thrown repeatedly till it falls “six” twice. What is the probability it takes 6 throws?

8. For dice are thrown. What is the probability that “five” is the highest number thrown and “three” is the lowest?

9. An infinite sequence of independent trials is to be performed. Each trial results in a success with probability p and a failure with probability q = 1 − p. What is the probability that

(a) At least 1 success occurs in the first n trials?

(b) Exactly k successes occur in the first n trials?

(c) All trials result in successes?

(d) The first trial occurs in the nth trial?

(e) It takes n trials to obtain 2 successes?

10. Suppose that each child born to a couple is equally likely to be a boy as a girl independent of the sex distribution of the other children in the family. For a couple having 5 children, compute the probabilities of the following events:

(a) all children are of the same sex,

(b) the 3 eldest are boys and the 2 youngest are girls,

(c) exactly 3 are boys,

(d) the two oldest are boys,

(e) there is at least 1 girl.

11. In successive rolls of a fair die, what is the probability of getting a seven before 3 even numbers?

For next class read section 3.1 and do problems 1, 2, 4, 5, 6, 8, 9, 10, 11, 12, 14, 15–21.