Question

Consider a game where each of 10 people randomly drop a dollar bill with their name on it into a bag and then take turns picking a dollar each from the bag.

a) What is the probability that at least one person picks a bill with his name on it?

b) Given that the first person to pick a bill with his name on it wins all the money, what are the chances of winning if you draw first? What is the best order to draw to increase your chances of winning?

Answer 1

Part (a)

Let be the events that the 1^st, 2^nd, ..., 10^th person picks the bill with his name on it respectively.

Then, the probability that at least one person picks a bill with his name on it is

Now, E_i occurs when i^th person chooses his own dollar, no matter what the others choose. Hence, the other dollars can be distributed in 9! ways. The total number of ways to distribute 10 dollars is 10!. Hence .

Again, E_i and E_j both can occur in 8! ways, as rest of the dollars can be distributed among 8 remaining persons in 8! ways. Hence,

Proceeding similarly,

Hence, required probability =

Part (b)

The person to draw the first dollar has 10 choices, of which he has only one way to win all the money, i.e. by choosing his own dollar.

Hence required probability =

Let us the consider the situation that (k - 1) persons has already drawn the dollar bills and none of them has got their own bills.

The k^th person will win only if all of the previous persons fail to get their own dollars and iff none of them drew his card and he selects his own card.

Hence, the required probability

Since this is a decreasing function in k, the probability will get increased if k is less.

Hence the 1st person will have highest chance of winning.