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There is a $50鈥�50$ chance that the queen carries the gene for hemophilia. If she is a carrier, then each prince has a $50鈥�50$ chance of having hemophilia. If the queen has had three princes without the disease, what is the probability that the queen is a carrier? If there is the fourth prince, what is the probability that he will have hemophilia?

On the morning of September $30,1982$, 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 $3$games remaining. All $3$of the Giants鈥� games were with the Dodgers, and the $3$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.

A town council of $7$ members contains a steering committee of size $3$. New ideas for legislation go first to the steering committee and then on to the council as a whole if at least $2$of the $3$committee members approve the legislation. Once at the full council, the legislation requires a majority vote (of at least $4$) to pass. Consider a new piece of legislation, and suppose that each town council member will approve it, independently, with probability p. What is the probability that a given steering committee member鈥檚 vote is decisive in the sense that if that person鈥檚 vote were reversed, then the final fate of the legislation would be reversed? What is the corresponding probability for a given council member not on the steering committee?

Suppose that each child born to a couple is equally likely to be a boy or a girl, independently 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 others girls.

(c) Exactly $3$are boys.

(d) The $2$oldest are girls.

(e) There is at least $1$girl.

$A$and $B$alternate rolling a pair of dice, stopping either when $A$rolls the sum $9$or when $B$rolls the sum $6$. Assuming that $A$ rolls first, find the probability that the final roll is made by $A$.

In a certain village, it is traditional for the eldest son (or the older son in a two-son family) and his wife to be responsible for taking care of his parents as they age. In recent years, however, the women of this village, not wanting that responsibility, have not looked favorably upon marrying an eldest son.

(a) If every family in the village has two children, what proportion of all sons are older sons?

(b) If every family in the village has three children, what proportion of all sons are eldest sons?

Assume that each child is, independently, equally likely to be either a boy or a girl.

Suppose that E and F are mutually exclusive events of an experiment. Suppose that E and F are mutually exclusive events of an experiment. Show that if independent trials of this experiment are performed, then E will occur before F with probability P(E)/[P(E) + P(F)].

Consider an unending sequence of independent trials, where each trial is equally likely to result in any of the outcomes $1,2or3,$. Given that outcome $3$is the last of the three outcomes to occur, 铿乶d the conditional probability that

(a) the 铿乺st trial results in outcome $1$;

(b) the 铿乺st two trials both result in outcome $1$.

A and B play a series of games. Each game is independently won by A with probability p and by B with probability $1$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 $4$games are played.

(b) Find the probability that A is the winner of the series

In successive rolls of a pair of fair dice, what is the probability of getting $2$sevens before $6$even numbers?