Ordered Cards
Goldman Sachs
You have a deck of 46 distinct cards. You shuffle the deck well and take the top 6 cards. If the cards are in strictly ascending or strictly descending order, you win $w$ amount, and you pay 1 to play the game. What should $w$ be for this game to be fair?
Answer
For this game to be fair, the expected net payoff of playing it should be 0, so that you neither win nor lose on average. This means the expected gain should equal the expected cost. So what is the expected payoff of playing the game?
If we win, we get $w$ and pay 1, whereas if we lose we only pay 1. So the payoff is $P(\text{win})\cdot (w-1) + P(\text{loss})\cdot (-1)$. So what is our probability of winning? We win when the 6 cards are in either ascending or descending order. We drew 6 random cards between 1 and 46, and we know they are all distinct. This means there's only one way in which they are in ascending order, and one way in which they are descending. There is a total of $6!$ ways to reorder the 6 cards, so our chance of winning is $\frac{2}{6!}$. Notice the fact that there are 46 cards is simply a distraction, it makes no difference to our probabilities - what matters is how many cards we draw and the fact that they are all distinct. Our chance of losing is then $1-P(\text{win})=\frac{6!-2}{6!}).
Putting all this together, we know the expected value of the game is:$$\frac{2}{6!}(w-1) + \frac{6!-2}{6!}(-1)=0$$So for this game to be fair, $w$ must be 360.