H S Hall and S R Knight Solutions for Chapter: Probability, Exercise 4: EXAMPLES. XXXII. d.

$A$ speaks the truth $2$ out of three times, and $B$ speaks the truth $4$ out of $5$ times, they agree in the assertion that from a bag containing $6$ balls of different colours a red ball has been drawn, find the probability that the statement is true.

From a pack of $52$ cards, one card has been lost, from the remaining pack of the cards, two cards are drawn and are found to be spades. Find the chance that the missing card is a spade.

There is a raffle with $10$ tickets and two prizes of value Rs.$50$ and Rs.$10$, respectively. A holds one ticket and is informed by $B$ that he has won the Rs.$50$ prize, while $C$ asserts that he has won the Rs.$10$ prize. What is $A^{\text{'}}$s expectation, if the credibility of $B$ is denoted by $\frac{2}{3}$ and that of $C$ is $\frac{3}{4}$?

A purse contains four coins, two coins having been drawn are found to be sovereigns, find the chance $\left(1\right)$ that all the coins are sovereign $\left(2\right)$ that if the coins are replaced, another drawing will give a sovereign.

From a bag containing $n$ balls, all either white or black, all numbers of each being equally likely, a ball is drawn which turns out to be white, this is replaced, another ball is drawn which also turn out to be white, if this ball is replaced, prove that the chance of the next draw giving a black ball is $\frac{1}{2}\left(n-1\right){\left(2n+1\right)}^{-1}$.

If $mn$ coins have been distributed to $m$ purses, $n$ into each find 

$\left(1\right)$ The chance that two specified coins will be found in the same purse, and 

$\left(2\right)$ What the chance becomes when $r$  purses have been examined and found not to contain either of the specified coins.

$A \text{and} B$ are two inaccurate arithmeticians whose chance of solving a given question correctly are $\frac{1}{8}$ and $\frac{1}{12}$, respectively. If they obtain the same result and if it is $1000$ to $1$ against their making the same mistake, find the chance that the result is correct.

Ten witnesses each of whom makes but one false statement in six, agree in asserting that a certain event took place. Show that the odds are five to one in favour of the truth of their statement, even although the priori probability of the event is as small as $\frac{1}{5^9+1}$.