To all you math geeks out there ... a statistics problem

Applying to College
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<p>Suppose you are on a game show with 28 other contestants. The game is for each contestant to go and pick 1 of 30 sheets of paper scattered on the floor. 5 randomly placed papers have a prize written on the back of them. (you also cannot see through the paper) The game show host picks you and asks whether you want to be the first to pick or the last. Which choice would you pick? Is it to your advantage to pick one choice over the other?</p>

<p>So, I've been thinking about this question. I recently won a prize in a group game by picking to be last. In the end, I knew there was 1 prize left, and I had to pick between two papers. I picked the right paper, and I won the prize. I think the right choice is to pick being last. However, I'm not sure how to statistically prove this. I have some ideas, and I seemed to have just proved that either choice has the same probability. However, I'm not confident on this answer.</p>

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<p>This isn’t necessarily a stats question; it’s just a ‘smartness’ question.
Solution 1: Short and sweet: Imagine a row of 30 papers. The probability that the 28 other people get some kind of permutation/combination ignores if you’re first or last. Therefore you can force the other 28 people into some sequence. So, if you’re first, prob that you win= prob that 28 get the specific combinations for the last 28 choices<em>prob that you actually get one of 2. If you’re last, prob that you win=probab that 28 get the specific combinations for the first 28 choices</em>prob that you get one of 2. So, they’re the same. </p>

<p>Solution 2: If you choose first, you get 5/30=1/6 chance of winning. For the last, you have 2 papers. You don’t know what they are, and “God” doesn’t know what they are either, so you must do it by casework:

  1. If both are bad, just ignore.
  2. One is good, one is bad. This happens if you have 4 good ones and 24 bad ones chosen. The number of times this happens is (5 choose 4)<em>(25 choose 24), and the total times this happens is (30 choose 28). Dividing, you’d get 25/87. Now the probability that you actually hit the prize is 1/2, so multiplying it by 1/2, you get 25/174.
  3. Both are good. This means that the rest chose 3 good ones, 25 bad ones. Again, this becomes (5 choose 3)</em>(25 choose 25), divided by (30 choose 28). Since you have a probability of 1 for choosing, it gives you 2/87.
    Adding up, you get 29/174, which is 1/6. They’re the same.</p>