Probability

In the game of Two-Finger Morra, 2 players show 1 or 2 fingers and simultaneously guess the number of fingers their opponent will show. If only one of the players guesses correctly he wins the amount (in dollars) equal to the sum of the fingers shown by him and by his opponent. If both players guess correctly or neither guesses correctly, then no money is exchanged. Consider a specified player and denote by X the amount of money he wins in a single game of Two-Finger Morra ... *(Please see attachment for complete question)

Probability

1. Suppose that two teams play a series of games that ends when one of the teams has one i number of games. Suppose that each game played is, independently, won by team A with probability p. Find the expected number of games that are played when (a) i = 2 and when (b) i = 3. Show also in both cases that this number is maximized when p = ½. 2. Consider problem 1 with i = 2. Find the variance of the number of games played and show that this number is maximized when p = ½.

Introductory Probability Course at the 400 Level.

Suppose that it takes at least 9 votes from a 12-member jury to convict a defendant. Suppose the probability that a juror votes a guilty person innocent is 0.2. whereas the probability that the juror votes an innocent person guilty is 0.1. If each juror acts independently and if 65% of the defendants are guilty, find the probability that the jury renders a correct decision. What % of defendants are convicted?

Introductory Probability Course at the 400 Level.

Two athletic teams play a series of games; the first team to win 4 games is declared the overall winner. Suppose that one of the teams is stronger than the other and wins each game with probability 0.6, independant of the outcomes of the other games. Find the probability that the stronger team wins the series in exactly i games. Do it for i=4,5,6,7. Compare the probability that the stronger team wins with the probability that it would win a 2-out-of-3 series. (See attachment for full question)

Probability (Ordered and Non-ordered Sets; Sampling; Replacing)

(Answer first if sampling with replacing, and then if sampling without replacing): Suppose that a batch of 100 items contains 6 that are defective and 94 that are nondefective. If X is the number of defective items in a randomly drawn sample of 10 items from the batch, find (a) P{X=0} and (b) P{X>2}