Friday Bonus Puzzle -- free tickets

Re: Friday Bonus Puzzle -- free tickets

Larry Barrett

The birthday problem comes into play here. That problem says that the 23th person in line has a (slightly) better than 50-50 chance of having a birthday the same as one of the 22 people before them in the line.

Your question is "what position should you take in line to maximize your chance of winning?". I think the 23rd position is still the correct answer, but not sure. If you look at the curve that represents the probability of having the same birthday as someone ahead of you in line, the slope of the curve is greatest at about this point, although I did not make any calculations.

Correction

Alex Y

The birthday problem comes into play here. That problem says that the 23th person in line has a (slightly) better than 50-50 chance of having a birthday the same as one of the 22 people before them in the line.

Larry, I think the problem you are thinking of is the point at which there is a 50-50 chance that there are two people in the room that have the same birthday (any two people, not just the last person and one other). The probability that the last person entering the room will have the same birthday as one of the other 22 people is 1-(364/365)^22=~6%.

But you are right that this problem is closely related to that one, and the answer is close to (but not equal to) 23.

Re: Correction

Larry Barrett

Alex, you are quite right.

To answer your problem, we need to calculate the probability that the nth person in line has the same birthday as one of the n-1 people ahead of him in line (which you already indicated), multiplied by the probability that none of the other people already in line had the same birthday as someone ahead of them.

When I do this computation, I get a result that slowly increases as n goes from 2 to 20, where this combined probability = .0318, then starts to decrease again. So I think the best place in line is position 20.

Re: Correction

David Weaver

If larry's answer is not correct, I would like to take a shot at this before the answer pops up.

Correct

Alex Y

Larry is correct that you want to be 20th in line.

To win, each person in front of you must NOT match someone in front of them, then you must match one of the ones in front of you. So, for instance, if you are the 20th person in line, the probabilities are as follows:

1st person in line: 1 (there is a 100% chance that this person will not match anyone in front of him or her)

2nd: 364/365 (you have a chance of winning only if his birthday is different from #1's)

3rd: 363/365 (his birthday must be different from the first two for you to have a chance of winning)

...

19th: 347/365 (his birthday is none of the 18 birthdays given by those earlier in line)

You in 20th position: 19/365 (the probability that your birthday is one of the 19 distinct dates given earlier.

Multiplying all those probabilities gives a probability of getting the prize of 3.232% (probably a cumulative rounding difference from what Larry stated--he may have been carrying more digits), which is the highest of any position.