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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?

Show that

$\frac{P(H鈭�E)}{P(G鈭�E)}=\frac{P\left(H\right)}{P\left(G\right)}\frac{P(E鈭�H)}{P(E鈭�G)}$

Suppose that, before new evidence is observed, the hypothesis $H$is three times as likely to be true as is the hypothesis $G$.

If the new evidence is twice as likely when $G$is true as it is when $H$is true, which hypothesis is more likely after the evidence has been observed?

A couple has $2$ children. What is the probability that both are girls if the older of the two is a girl ?

Let $A,B$, and $C$be events relating to the experiment of rolling a pair of dice.

(a) If localid="1647938016434" $P\left(A\right|C)>P(B\left|C\right)$and localid="1647938126689" $P\left(A\right|C^c)>P(B|C^c)$either prove that localid="1647938033174" $P\left(A\right)>P\left(B\right)$or give a counterexample by defining events $B$and $C$for which that relationship is not true.

(b) If localid="1647938162035" $P\left(A\right|C)>P(A|C^c)$and $P\left(B\right|C)>P(B|C^c)$either prove that $P\left(AB\right|C)>P(AB\left|C^c\right)$or give a counterexample by defining events $A,B$and $C$for which that relationship is not true. Hint: Let $C$be the event that the sum of a pair of dice is $10$; let $A$be the event that the first die lands on $6$; let $B$be the event that the second die lands on $6$.

An investor owns shares in a stock whose present value is $25$.She has decided that she must sell her stock if it goes either down to 10 or up to $40$. If each change of price is either up $1$point with probability .$55$or down $1$point with probability .$45$, and the successive changes are independent, what is the probability that the investor retires a winner?

Consider$3$urns. An urn $A$contains$2$white and $4$red balls, an urn $B$contains $8$white and 4 red balls and urn $C$contains $1$white and $3$red balls. If $1$ball is selected from each urn, what is the probability that the ball chosen from urn $A$was white given that exactly $2$white balls were selected?