A game involves you choosing one number (between 1 to 6 inclusive) and then throwing three fair dice simultaneously. If none of the dice shows up the number that you have chosen, you lose Rs. 1. If exactly one, two or three dice show up the number that you have chosen, you win Rs. 1, Rs. 3 or Rs. 5 respectively. What is your expected gain?
Please suggest some other questions of this kind, if you have any.
Say you choose $6$. The chance you get exactly two sixes is ${3 \choose 2}$ ways to choose the dice that come up six times $(\frac 16)^2$ that they come up six times $\frac 56$ that the other comes up something else. Do the same calculation for each number of sixes. Multiply each probability by the return of that result and add.