Question

There is a raffle with $12$ tickets and two prizes of $9$ pounds and $3$ pounds A, B, C, whose probability of speaking the truth are $\frac{1}{2},\frac{2}{3},\frac{3}{5}$ respectively, report the result to D, who holds one ticket. A and B assert that he has won the $9$ pounds prize, and C asserts that he has won the $3$ pounds prize; what is D's expectation?

Answer 1

Three cases are possible; D may have won $9$pounds, $3$ pounds, or nothing, for A, B, C may all have spoken falsely.
Now, we have the priori probabilities
$P_1=\frac{1}{12},P_2=\frac{1}{12},P_3=\frac{10}{12}$;
also $p_1=\frac{1}{2}\times \frac{2}{3}\times \frac{2}{5}=\frac{4}{30},p_2=\frac{1}{2}\times \frac{1}{3}\times \frac{3}{5}=\frac{3}{30},p_3=\frac{1}{2}\times \frac{1}{3}\times \frac{2}{5}=\frac{2}{30};$
$\therefore \frac{Q_1}{4}=\frac{Q_2}{3}=\frac{Q_3}{20}=\frac{1}{27};$
Hence D's expectation $=\frac{4}{27}$ of $9$ pounds $+\frac{3}{27}$ of $3$ pounds $=1$ pounds. $13s.4d$.