Question

$P$ makes a bet with $Q$ of $8$ pounds to $120$ pounds that three races will be won by the three horses $A,B,C,$ against which the betting is $3$ to $2,4$ to $1,$ and $2$ to $1$ respectively. The first race having been a on by $A$, and it being known that the second race was won either by $B$, or by a horse $D$ against which the betting was $2$ to 1, find the value of $P^{\prime }s$ expectation.

Answer 1

At first, B's chance of winning his race is $\frac{1}{5}$; similarly C's chance is $\frac{1}{3}$ and D's chance is $\frac{1}{3}$

$\therefore$ after the second race is known to have been won by B or D

B's chance:certainty$=\frac{1}{5}.\frac{1}{3}=3.5$

B's chance$=\frac{3}{8}$

Chance of P winning his bet$=1\times \frac{3}{8}\times \frac{1}{3}=\frac{1}{8}$

and the chance of P losing it is $\frac{7}{8}$

Thus P's expectation$=\frac{1}{8}$ of pounds $120-\frac{7}{8}$ pounds $8$

$=8$ pounds