Question

A speaks truth $3$ times out of $4$, and B speaks $7$ times out of $10;$ they both assert that a white ball has been drawn from a bag containing $6$ balls all of different colours. Find the probability of the truth of the assertion.

Answer 1

There are two hypothesis; (i) their coincident testimony is true, (ii) it is false.
Here $P_1=\frac{1}{6},P_2=\frac{5}{6}$;
$p_1=\frac{3}{4}\times \frac{7}{10},p_2=\frac{1}{25}\times \frac{1}{4}\times \frac{3}{10};$
For in estimating $p_2$ we msut take into account the chance that A and B will both select the white ball when it has not been drawn; this chance is $\frac{1}{5}\times \frac{1}{5}$ or $\frac{1}{25}$.
Now the probabilities of the two hypotheses are as $P_1{p}_1$ to $P_2{p}_2$, and therefore as $35$ to $1$; thus the probability that the statement is true is $\frac{35}{36}$.