Russian Roulette

Q. What's the probability of winning (surviving) a game of Russian Roulette by volunteering to go first (pull the trigger with gun at your head) where each player spins the barrel before pulling the trigger ?

A. Let x be the probability of surviving by going first. The probability of surviving going first is 5/6 multiplied by the probability that the other player loses (which is 1-x because the other player faces the same game).

$$ \begin{eqnarray} x = \frac{5}{6} (1-x) \\\ x = \frac{5}{11} \end{eqnarray} $$

So a little less than half!

Dice throws

Q. What's the fair value of a game where you get x$ for a throw of a dice where x is the number (1-6) that results from the throw of the dice ?

A. 3.5 which is the expected value (mean) of the numbers 1-6.

Q. What's the fair value of a game where you have the option of throwing again if you don't like your first throw, but have to accept the outcome of the second throw ?

A. You're only going to go for a second throw if your first throws up < 3.5. You'll keep the result of your first throw if it's 4,5 or 6 (probability of which is 0.5). You're only going to go for a second throw if your first throws up < 3.5 which also has probability 0.5. The expected value of the second throw still remains 3.5.

$$ \begin{eqnarray} x = \frac{1}{2} (4+5+6)/3 + \frac{1}{2} (3.5) \\\ x = \frac{1}{2} (5) + \frac{1}{2} (3.5) \\\ x = 4.25 \end{eqnarray} $$