In a game, a person is paid Rs. 2 if he gets all heads or all tails when three coins are tossed, and he will pay Rs. 2 if either one or two heads show. What can he expect to win on an average per game?

When three coins are tossed, there are 8 possible outcomes.

The person wins Rs. 2 when all three coins show heads (HHH) or all three show tails (TTT).

The person loses Rs. 2 when exactly 1 head or exactly 2 heads appear.

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The 8 possible outcomes are:

HHH - All heads → Win Rs. 2

HHT - 2 heads → Lose Rs. 2

HTH - 2 heads → Lose Rs. 2

THH - 2 heads → Lose Rs. 2

HTT - 1 head → Lose Rs. 2

THT - 1 head → Lose Rs. 2

TTH - 1 head → Lose Rs. 2

TTT - All tails → Win Rs. 2

Winning outcomes: 2 (HHH and TTT)

Losing outcomes: 6 (all others)

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Probability of winning $= \dfrac{2}{8} = \dfrac{1}{4}$

Probability of losing $= \dfrac{6}{8} = \dfrac{3}{4}$

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Expected Value $= (\text{Probability of winning} \times \text{Amount won}) + (\text{Probability of losing} \times \text{Amount lost})$

Expected Value $= \left(\dfrac{1}{4} \times 2\right) + \left(\dfrac{3}{4} \times (-2)\right)$

Expected Value $= \dfrac{2}{4} + \dfrac{-6}{4}$

Expected Value $= \dfrac{2}{4} - \dfrac{6}{4}$

Expected Value $= \dfrac{-4}{4}$

Expected Value $= -1$

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The expected value is Rs. $-1$ per game.

The negative value indicates a loss of Rs. 1 on average per game.