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?
Comments on this teaser
|Posted by Paladin||04/23/09|
|Probabilities always make me stop and think. I haven't had any probability courses, but I can usually muddle through with the math I do know and get a decent shot at it. This was fun but very challenging. Well done!|
|Posted by Shadows||04/23/09|
|Well . . . I think I'll just look on the bright side. I got the first one! :lol:
I see you put quite some work into this. It's a nice teaser. When I saw it in proofreading, I rated it fun and hard.
|Posted by bradon182001||04/30/09|
|Waaaaay over my head.Excellent teaser. Thanks for posting. :o|
|Posted by gooberbaby1||06/11/10|
|I would have just put down an answer,NOT an answer & a HUGE explanation :oops: :oops:|
|Posted by racoonieboy||07/25/10|
|Actually, the explanation... well, explained a lot and I like it!|
|Posted by Zag24||08/04/10|
|I'm betting, goober, that you wouldn't have put the answer at all, since I bet you couldn't have come up with it.|
|Posted by Jaypuzzle12||07/21/11|
|THe answer was well explained. Good job! :D|
|Posted by ali_p||09/13/12|
|I don't think this is right...what accounts for the fact that you're more likely to match 2 games when you have 5 games per ticket instead of just having to match one game to another? I think you're using the wrong "P"...otherwise, what would account for it being less likely to match 3 out of 5 versus just 2 out of 5??|
|Posted by Zag24||02/15/13|
|all_p, sorry I didn't see your question until just now.
In the explanation, step 2, there are four fractions we are multiplying:
(5245785 * 5245784 * 5245783 * 5245782) / 5245786 ^ 4
That is, we are confirming that all 5 combos on the ticket are different. If we had only, say, three combos on one ticket, that step would only be
(5245785 * 5245784) / 5245786 ^ 2
and if there were only 2 combos on one ticket, it would be the same as the first step,
5245785 / 5245786
Remember that these are the chances that all the combos are different. The chances that there are two the same is 1 minus this.|
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