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Chapter 3: Q. 3.25 (page 109)
Prove directly that,
By applying the definition of conditional probability,the direction starting from the right side.
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In a class, there are 4 first-year boys, 6 first-year girls, and 6 sophomore boys. How many sophomore girls must be present if sex and class are to be independent when a student is selected at random?
Suppose that an ordinary deck of 52 cards (which contains 4 aces) is randomly divided into 4 hands of 13 cards each. We are interested in determining p, the probability that each hand has an ace. Let Ei be the event that I the hand has exactly one ace. Determine p = P(E1E2E3E4) by using the multiplication rule.
Three prisoners are informed by their jailer that one of them has been chosen at random to be executed and the other two are to be freed. Prisoner A asks the jailer to tell him privately which of his fellow prisoners will be set free, claiming that there would be no harm in divulging this information because he already knows that at least one of the two will go free. The jailer refuses to answer the question, pointing out that if A knew which of his fellow prisoners were to be set free, then his own probability of being executed would rise from 1 3 to 1 2 because he would then be one of two prisoners. What do you think of the jailer鈥檚 reasoning?
A and B play a series of games. Each game is independently won by A with probability p and by B with probability 鈭� p. They stop when the total number of wins of one of the players is two greater than that of the other player. The player with the greater number of total wins is declared the winner of the series.
(a) Find the probability that a total of games are played.
(b) Find the probability that A is the winner of the series
On the morning of September , the won鈥搇ost records of the three leading baseball teams in the Western Division of the National League were as follows:
Each team had games remaining. All of the Giants鈥� games were with the Dodgers, and the remaining games of the Braves were against the San Diego Padres. Suppose that the outcomes of all remaining games are independent and each game is equally likely to be won by either participant. For each team, what is the probability that it will win the division title? If two teams tie for first place, they have a playoff game, which each team has an equal chance of winning.
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