MHTCET 23 Apr 2025 Shift 1 Paper

Concept:Expected value is the sum of each possible net gain/loss multiplied by its probability.Explanation:Probability of getting 5 or 6 on a fair die is $\frac{2}{6} = \frac{1}{3}$.Probability of not getting 5 or 6 is $1 - \frac{1}{3} = \frac{2}{3}$.Let $X$ be the net gain in rupees.Case 1: Wins on the first throw: gain $= 40$, probability $= \frac{1}{3}$.Case 2: Loses the first, wins the second: net $= -20 + 40 = 20$, probability $= \frac{2}{3} \times \frac{1}{3} = \frac{2}{9}$.Case 3: Loses the first two, wins the third: net $= -20 - 20 + 40 = 0$, probability $= \frac{2}{3} \times \frac{2}{3} \times \frac{1}{3} = \frac{4}{27}$.Case 4: Loses all three throws: net $= -60$, probability $= \left(\frac{2}{3}\right)^3 = \frac{8}{27}$.Expected value:$$E(X) = 40\left(\frac{1}{3}\right) + 20\left(\frac{2}{9}\right) + 0\left(\frac{4}{27}\right) + (-60)\left(\frac{8}{27}\right)$$$$= \frac{360 + 120 - 480}{27} = \frac{0}{27} = 0$$Thus, the expected gain/loss is ₹0.Answer:The expected gain/loss is $0$ rupees, i.e., option C.