Cub11k's BIU Notes

Probability 03

15 min read

Exercise 0.1

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Exercise 0.2

Exercise 0.3

A gambler plays independent rounds of a casino game. In each round he wins with probability and loses with probability . He will definitely play at least 3 rounds. After that, if he accumulates a total of 2 wins (including wins he got in the first 3 rounds) before the 5th round, he stops playing immediately. Otherwise, he stops playing after the 5th round. For example, if he won in the first 2 rounds, he also play the third round and then he quits. If he lost the first round and won round 2 and round 3, then he quits immediately after round 3. If he won once or less in the first 3 rounds, he continues to play until he accumulates a total of 2 wins, or after the 5th round, whatever comes first.

  1. Express the number of rounds in terms of Geometric random variables, and max, min operators.
  2. Find the probability mass function of the number of rounds.
  3. What is the expectation and variance of the number of rounds?

Exercise 0.4

Five distinct numbers are randomly distributed to players numbered 1 through 5. Whenever two players compare their numbers, the one with the higher one is declared the winner. Initially, players 1 and 2 compare their numbers; the winner then compares her number with that of player 3, and so on. Let denote the number of times player 1 is a winner. Find .

Exercise 0.5

Exercise 0.6

State your assumptions. Suppose that the average number of cars abandoned weekly on a certain highway is 2.2. Approximate the probability that there will be

  • (a) no abandoned cars in the next week;
  • (b) at least 2 abandoned cars in the next week.

Exercise 0.7

A factory produces electronic components in a month. Each component is defective with probability , independently of the others. Let denote the number of defective components produced in a given month.

  • (a) Find the probability that exactly 3 defective components are produced
  • (b) Find probability that at least 2 defective components are produced

Exercise 0.8

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Independent trials are performed, each with probability of success. Let denote the number of trials needed to obtain exactly successes. Compute the expectation and variance of . (Hint: can be represented as a sum).

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