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, and C against which the betting is 3 to 2, 4 to 1, 2 to 1 respectively. The first race having been won by A and it is known that the second race was won either by B or by a horse C against which the betting was 2 to 1, find the value of P’s expectation?

Answer 1

Hint: To find P’s expectation, we have to, first of all, find the chance of winning the second race by horse B and the third race by C and also when the second race is won by horse C and the third race by horse B. P will win the bet when the second race will be won by horse B and the third race will be won by horse C and also P will lose if the second race will be won by horse C and the third race will be won by horse B. The P’s expectation is equal to the multiplication of chance of winning of P with 120 and subtracting the multiplication of chance of losing of P with 8.

Complete step by step answer:
Probability, when horse B wins, is equal to:
 $ \dfrac{1}{5} $
Probability, when horse C wins, is equal to:
 $ \dfrac{1}{3} $
Now, P will win the bet when horse B wins the second race, and horse C will win the third race.
The ratio of B’s chance to win and certainty that B will win the second race is equal to:
The certainty that either the second or third race will win by horse B or horse C is equal to:
 $ \dfrac{1}{3} $
So, the ratio of B’s chance to win and the certainty is equal to the ratio of the probability of B winning the race to the probability of certainty that either the second or third race will win by horse B or horse C.
 $ \dfrac{1}{5}:\dfrac{1}{3} $
Now, rearranging the above ratio we get,
 $ 3:5 $
Hence, the probability that horse B will win the second race is equal to:
 $ \dfrac{3}{8} $
Now, the chance of P winning the bet is equal to the multiplication of the probability of winning horse A in the first race followed by the probability of winning horse B in the second race, and the probability of winning horse C in the third race.
In the above, it is already given that first race is won by horse A so its probability will be 1 and the winning probability of horse B in the second race we have shown above and winning probability of horse C we have shown above as $ \dfrac{1}{3} $ .
Hence, chance of P winning the bet is equal to:
 $ 1\times \left( \dfrac{3}{8} \right)\times \left( \dfrac{1}{3} \right) $
3 will be cancelled out from the numerator and the denominator and we are left with:
 $ \dfrac{1}{8} $
Now, probability of P losing the bet is equal to:
 $ \begin{align}
  & 1-\dfrac{1}{8} \\
 & =\dfrac{8-1}{8} \\
 & =\dfrac{7}{8} \\
\end{align} $
Hence, expectation of P winning the bet is equal to the multiplication of probability of P winning the bet with 120 and then subtracting the result of this multiplication with the result of multiplication of P losing the bet with 8 we get,
 $ \begin{align}
  & \dfrac{1}{8}\times 120-\dfrac{7}{8}\times 8 \\
 & =15-7=8 \\
\end{align} $
Hence, the expectation of P winning the bet is 8.

Note:
 Whenever it is given that chance of winning against A is 3 to 2. This means that the chances of winning A are $ \dfrac{2}{5} $ which is equal to 2 divided by the addition of 3 and 2. You can remember it by the word “against”, the number which is written just after the word “against” in the ratio is the number which is against the winning and the number which is written after the ratio sign is the number which is in favor of winning. Like, 3 is the number that indicates a chance of losing A, and 2 indicates a chance of winning A.