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I was talking with a friend and we were discussing a math problem disguised as a social situation:

If the chance that someone to accept your request to go out with them was 1% and you asked 1 person only, then the probability they would say yes is .01

If you were to ask 100 people, then your chances are still .01 according to my friend.

Now my question is, why does it seem intuitive that the chances would increase when asking 100 people rather than 1? Is it the expected value that's causing this intuition?

Can this situation be modeled using a balls & bins problem?

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up vote 5 down vote accepted

The problem stems from ambiguous use of term chance and probability. Probability can be defined as a ratio of successful outcomes and number of trials. So if we increase the number of trials and let the probability remain the same, the number of successful outcomes must increase in order that the ratio remains the same. So if we say that number of successful outcomes are chances, then yes, chances are increasing, yet the probability remains the same.

Another way to look at that is to look at the probability of getting a no. With one person the probability is $0.99$. If you ask 100 persons, then the probability that they all say no is $0.99^{100}\approx 0.36$. So the probability that you will get at least one positive answer after asking 100 persons is actualy $1-0.99^{100}\approx0.64$. If you ask 1000 persons, then the probability of getting a no from all of them is $0.99^{1000}\approx 4.31\times 10^{-5}$, so probability of one yes is $1-0.99^{1000}\approx0.99999$. But that is not surprising, since saying that probability of a yes is 0.01 you are effectively saying that 1% of population will say yes. So among 1000 persons there should be 10 who say yes, so getting only one yes from 1000 persons is less likely (the event has smaller probability) than getting 10.

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Excellent. Thank you! – Sev Dec 15 '10 at 20:47

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