A Bayesian procedure for the sequential estimation of the by Marcus R.

By Marcus R.

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Two cards are randomly selected from a deck of 52 playing cards. (a) What is the probability they constitute a pair (that is, that they are of the same denomination)? (b) What is the conditional probability they constitute a pair given that they are of different suits? 26. A deck of 52 playing cards, containing all 4 aces, is randomly divided into 4 piles of 13 cards each. Define events E1 , E2 , E3 , and E4 as follows: E1 = {the first pile has exactly 1 ace}, E2 = {the second pile has exactly 1 ace}, E3 = {the third pile has exactly 1 ace}, E4 = {the fourth pile has exactly 1 ace} Use Exercise 23 to find P (E1 E2 E3 E4 ), the probability that each pile has an ace.

If the sum is anything else, then she continues throwing until she either throws that number again (in which case she wins) or she throws a seven (in which case she loses). Calculate the probability that the player wins. Exercises 17 14. The probability of winning on a single toss of the dice is p. A starts, and if he fails, he passes the dice to B, who then attempts to win on her toss. They continue tossing the dice back and forth until one of them wins. What are their respective probabilities of winning?

11. If two fair dice are tossed, what is the probability that the sum is i, i = 2, 3, . . , 12? 12. Let E and F be mutually exclusive events in the sample space of an experiment. Suppose that the experiment is repeated until either event E or event F occurs. What does the sample space of this new super experiment look like? Show that the probability that event E occurs before event F is P (E)/ [P (E) + P (F )]. Hint: Argue that the probability that the original experiment is performed n times and E appears on the nth time is P (E) Ɨ (1 āˆ’ p)nāˆ’1 , n = 1, 2, .

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