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Suppose you want to buy a toy for your daughter and you want to make sure she will like it. To make sure she'll like it, you decide to ask her friends (for this problem you can imagine she has infinity friends). Each friend, independently, can tell you whether she will like it or not correctly with probability 2/3 (each friend gives a boolean answer though).

How can you find out your daughter will like the toy you are about to buy with probability >= 1 - 2^(1/n) where n is the number of friends you will ask?

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Suppose you ask 100 friends, and you get 65 votes saying yes and 35 for no. What's the obvious thing to do now? Can you generalise your answer to my previous question? – Srivatsan Nov 18 '11 at 4:47
$1-2^{1/n}$ is negative for every positive $n$. – Did Nov 18 '11 at 6:09
I assume that you meant probability at least as large as $1-2^{-n}$ not $1-2^{1/n}$ as you have it. But suppose that the toy you are about to buy (meaning that you have already made the decision which one you are going to buy) is in fact one that your daughter will not like. You want to know how many friends you should ask to be able to find out with probability $1-2^{-n}$ or more that your daughter will like your choice? – Dilip Sarwate Nov 18 '11 at 15:32

This question has no meaningful answer. Asking her friends whether she will like the toy can only update your prior estimate of the probability of her liking it; it cannot give you a probability out of nothing. For instance, if you have a terrible track record of always consistently coming up with bad ideas for presents and the a priori probability that this time you had a good idea is low, then you'll have to ask quite a number of friends until their positive answers might sufficiently convince you to buy the toy. The situation would be quite different if you have a lucky hand at choosing presents. Also, your daughter might be very picky and have a low a priori probability of liking any randomly chosen present, or she might enjoy presents in general and only dislike a few particular things. Without any information on these prior probabilities, it's impossible to answer the question.

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