favorite probability problems

[karlson]

I have to teach a section tomorrow dealing with some basic probability. I was wondering if you guys had some favorite fun problems to share. I won't be able to use anything super complicated, just things like computing expectations, variances, using bayes theorem, using common distributions, etc. The trick is just to find a cool context. I don't feel like using gambling examples much though.

I was going through some books when I realized that this may be the best resource.

Thanks guys.

[FrankLu99]

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I don't feel like using gambling examples much though.

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Good luck!

[Leo99]

I always remember the colored balls in the bag problems. 5 red balls, 3 blue balls. What the p you pull out a blue bal? What's the p you pull out 2 blue balls? What's the p you pull out 2 balls and exactly one is blue? What's the p you pull out 2 balls and at least one is blue? Pull out a ball, look at it, put it back in the bag. Do that 100 times. How many blue balls would you expect to see?

I don't know of any cooler contexts than that [img]/images/graemlins/crazy.gif[/img]

[Gator]

Here’s the proposition. You have 1,000 identical slips of paper. You write a distinct number on each piece of paper. There is no bound. The only limitations are that each number be unique and that a person of high school intelligence can quickly determine that one number is higher then another (i.e. 1.5345 is permissable as is 9,999,999.02 as is - 53 -- but i-squared is not). These 1,000 slips of paper are then mxed up in a hat. You randomly select a number and read it out loud. After reading a number, I either tell you to stop or go ahead and read the next number. I win if I stop you after you've read the highest number (before reading another number). I lose if you read the highest number and I tell you to go to the next number. I am only allowed to stop you one time. What percentage of the time will I win this proposition using an optimal strategy (and what is that strategy). There are no tricks (i.e. reading body language, marking the cards, etc.).

[karlson]

Yeah, this is a good one. I always knew it as the "best prize problem" or the "most beatiful woman" problem. (prizes/women are being displayed to you and you have to pick the best one).

[fnord_too]

[ QUOTE ]
Here’s the proposition. You have 1,000 identical slips of paper. You write a distinct number on each piece of paper. There is no bound. The only limitations are that each number be unique and that a person of high school intelligence can quickly determine that one number is higher then another (i.e. 1.5345 is permissable as is 9,999,999.02 as is - 53 -- but i-squared is not). These 1,000 slips of paper are then mxed up in a hat. You randomly select a number and read it out loud. After reading a number, I either tell you to stop or go ahead and read the next number. I win if I stop you after you've read the highest number (before reading another number). I lose if you read the highest number and I tell you to go to the next number. I am only allowed to stop you one time. What percentage of the time will I win this proposition using an optimal strategy (and what is that strategy). There are no tricks (i.e. reading body language, marking the cards, etc.).

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Isn't the optimal strategy here to go through the first 1/e slips, then stop the person on the first number than the highest in those 1/e? I forget what the chance of winning is, I just remember thinking "Damn e shows up everywhere!" (your chances of winning might be 1/e, too. Must be a conspiracy!)

[Gator]

The answer I know (and there could be a slightly more optimal strategy) is to let 1/2 the numbers get called and then pick as soon as a higher number is read. This leads to success better then 25% of the time. Second highest number in first half 50% - Highest number in second half 50% -- both events occurring (i.e. guaranteed win) 25%. You can pick up some wins even if it doesn't happen just like that so answer is > 25%.

[karlson]

fnord had it right - you can do better by letting N/e of the prizes go by and then picking the first one that's better than any of the others. In fact, N/e is optimal for the strategies of that type. It also seems obvious that these are the only type of strategies to consider.
I don't feel like typing the proof that N/e is optimal, but I'm sure you can find it if you feel like it.