Question

You are holding a party at home, and everyone is about to participate in the following game.

Each person will write their name on a card. All the cards will then be collected and randomly redistributed (one per person). If anyone gets back the card with their own name then all the cards will be collected and randomly distributed again, and this process will be repeated until no-one is holding the card with their own name.

When everyone has a card with someone else’s name on it, you will call out the name on your card. The called person will then call out the name on their card, and so on, until finally your own name is called out.

If anyone’s name does not get called out at some stage during this game, they will have to drink a whole 1 litre bottle of vodka by midnight.

(a) Suppose that there are five people at your party (including yourself).

(i)Find the probability that no one will have to drink 1 litre of vodka by midnight.

(ii)Then find the expected number of people who will have to drink 1 litre of vodka by midnight.

(b) Derive general formulas for the probability and expectation in (a), ones which are correct for any number of people attending your party (i.e. 2, 3, 4, etc.).

(c)Then apply these two formulas to the cases where there are 2, 3, 4, 5, 10 and 100 people at your party, respectively. Present your results in a table.

(very appreciated write in details, thank you very much)

Answer 1