In a game a man wins ₹ 40 if he gets 5 or 6 on a throw of a fair die and loses ₹ 20 for getting any other number on the die. If he decides to throw the die either till he gets a five or six or to a maximum of three throws, then his expected gain/loss (in rupees) is

Step-by-step solution

Let S denote success (5 or 6) and F denote failure (1, 2, 3, or 4). The probabilities are P(S) = 1 3 and P(F) = 2 3 , with gains of ₹40 and –₹20 respectively. The process terminates at the first success or after three throws. The four possible outcomes are: Success on first throw: P = 1 3 , net gain = ₹40 Failure then success: P = 2 3 1 3 = 2 9 , net gain = –20 + 40 = ₹20 Two failures then success: P = 2 3 2 3 1 3 = 4 27 , net gain = –20 – 20 + 40 = ₹0 Three failures: P = ( 2 3 )^3 = 8 27 , net gain = –20 – 20 – 20