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More ways to get Braingle...

Duplicate Lottery Picks

Probability puzzles require you to weigh all the possibilities and pick the most likely outcome.

 

Puzzle ID:#44355
Fun:** (1.85)
Difficulty:*** (2.95)
Category:Probability
Submitted By:Zag24Aus****!!
Corrected By:Zag24

 

 

 



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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Comments

PaladinAus*
Apr 23, 2009

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!
ShadowsAca*
Apr 23, 2009

Well . . . I think I'll just look on the bright side. I got the first one!

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.

Good job!
bradon182001*us*
Apr 30, 2009

Waaaaay over my head.Excellent teaser. Thanks for posting.
gooberbaby1*us*
Jun 11, 2010

I would have just put down an answer,NOT an answer & a HUGE explanation
racoonieboytus*
Jul 25, 2010

Actually, the explanation... well, explained a lot and I like it!
Zag24Aus*
Aug 04, 2010

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.
Jaypuzzle12
Jul 21, 2011

THe answer was well explained. Good job!
ali_p
Sep 13, 2012

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??
Zag24Aus*
Feb 15, 2013

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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