In a game, a man wins Rs. 100 if he gets 5 or 6 on a throw of a fair die and loses Rs. 50 for getting any other number on the die. If he decides to throw the die either till he gets a five or a six or to a maximum of three throws, then his expected gain/loss (in rupees)

Let X be the random variable representing the gain or loss in the game.

The probability of getting 5 or 6 on a throw is P(5 or 6) = 2/6 = 1/3.
The probability of getting any other number is P(other) = 4/6 = 2/3.

Case 1: The man gets 5 or 6 on the first throw.
The probability is 1/3. The gain is Rs. 100.
Expected gain = (1/3) * 100 = 100/3

Case 2: The man does not get 5 or 6 on the first throw, but gets 5 or 6 on the second throw.
The probability is (2/3) * (1/3) = 2/9. The gain is 100 - 50 = 50.
Expected gain = (2/9) * 50 = 100/9

Case 3: The man does not get 5 or 6 on the first two throws, but gets 5 or 6 on the third throw.
The probability is (2/3) * (2/3) * (1/3) = 4/27. The gain is 100 - 50 - 50 = 0.
Expected gain = (4/27) * 0 = 0

Case 4: The man does not get 5 or 6 in three throws.
The probability is (2/3) * (2/3) * (2/3) = 8/27. The loss is Rs. 150.
Expected loss = (8/27) * (-150) = -400/9

Total expected gain/loss = (100/3) + (100/9) + 0 + (-400/9) = (300 + 100 - 400) / 9 = 0

Therefore, the expected gain/loss is 0.