Duplicate Lottery Picks
In the Massachusetts Megabucks lottery, six different numbers from 1 to 42 (inclusive) are selected. When you buy a ticket, you can ask for a "quick pick" in which the computer chooses the numbers for you, and you can purchase up to five games on a single ticket. We'll assume that the computer's random number generator is fair, giving each possible combination an equal probability of being chosen.
1. If I "quick pick" for two games, what are the chances that the two games have the same combination of numbers?
2. If I "quick pick" for five games (one five-game ticket), what are the chances that there are two games on that ticket with the same combination?
3 (The toughie). How many five-game quick-pick tickets would I have to buy in order to have a greater than 50% chance of having at least one ticket with two games on it that match exactly?
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