Question

In a game, a person is paid Rs.$5$, if he gets all heads or all tails when three coins are tossed and he will pay Rs.$3$, if either one or two heads show. What can he expect to win on the average per game?

Answer 1

First find the possible outcomes in the toss of three coins and then find the probability of getting all heads and all tails. After that use the given amount according to the condition to get the desired result.

Complete step-by-step answer:
It is given in the problem that a person gets paid rupees 5 if he gets all heads and all tails when three coins are tossed and he has to pay rupees 3 if he gets one or two heads.
We have to find the expected amount to win on the average per game.
We have given that there are three coins tossed then the sample space of tossing three-coin is given as:
Sample space = \left\\{ {HHH,HHT,HTH,HTT,THH,THT,TTH,TTT} \right\\}
So, there are possibly eight outcomes on the toss of three coins.
Suppose that $A$be the event of getting all head or all tail and $B$be the event of getting one or two heads.
It is given in the problem that a person gets paid rupees 5 if he gets all heads and all tails, so there are two events of either getting head or tail. So, the probability of getting all head or all tail is given as:
Probability of getting all head or all tail,$P\left( A \right) = \dfrac{2}{8} = \dfrac{1}{4}$
Then the amount paid to a person when an event $A$occurs is given as$\left( {5 \times \dfrac{1}{4}} \right)$.
It is also given that the person has to pay rupees 3 if he gets one or two heads, then the probability of getting one or two heads is given as:
Probability of getting one or two heads,$P\left( B \right) = \dfrac{6}{8} = \dfrac{3}{4}$
Then the amount paid by a person when an event $B$occurs is given as$\left( { - 3 \times \dfrac{3}{4}} \right)$.
So, the expected amount to win on the average per game is given as:
Average amount$= \left( {{\text{Amount of winning}}} \right) + \left( {{\text{Amount of losing}}} \right)$
Average amount $= \left( {\dfrac{5}{4}} \right) + \left( { - \dfrac{9}{4}} \right)$
Average amount $= \dfrac{{ - 4}}{4} = - 1$
Thus, on the toss of three coins, the person will lose Rupees 1 on average.

Note: A sample space is the set of all possible outcomes or results for any experiment. In the given problem, three coins are tossed then there are two outcomes in every coin, one is head and the other is tails, and the total possible outcomes are 8.