Raju and Rani are playing betting game. Raju rolls a dice and if 5 comes on face then they stop the game, else Raju needs to pay 1 rupee to Rani and continue rolling. Expected no of rupees Rani gets in this game?

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1 Answer

No rupees are paid out if 5 comes up on the first throw. There is a 1/6 probability for this to occur.

1 rupee is paid out if 5 comes up on the second throw. The probability is 5/6×1/6=5/36.

2 are paid out if 5 comes up on the third throw. Probability is (5/6)²(1/6)=25/216.

n rupees are paid out if 5 comes up on the (n+1)th throw, probability=(5/6)ⁿ(1/6).

The sum of all probabilities is (1/6)(1+5/6+(5/6)²+...+(5/6)ⁿ)=(1/6)(1-(5/6)ⁿ⁺¹)/(1-(5/6))=1-(5/6)ⁿ⁺¹. This equals 1 as n➝∞. The mean of this distribution is when (5/6)ⁿ⁺¹=1/2, so n=ln(1/2)/ln(5/6)-1=2.80 approx. Half the data lie to the left of the mean and half to the right.

So since n is the payout, the average payout is 2.80 rupees.

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