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Old 04-14-2012, 10:11 PM
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yaser yaser is offline
Caltech
 
Join Date: Aug 2009
Location: Pasadena, California, USA
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Default Re: question about probability

Quote:
Originally Posted by pventurelli@gotoibr.com View Post
In the second lecture, the Professor asked a question about flipping a coin:
What is the probability of getting all ten heads if you flip a coin 10 times count the number of heads, and then you repeat the experiment 1000 times.

The answer he gave was 63% -- I would like to know how this was computed.

Any help would be greatly appreciated.

Thanks!
The probability of getting 10 heads for one coin is {1 \over 2} \times {1 \over 2} \times \cdots \times {1 \over 2} (10 times) which is aprroximately {1 \over 1000}.

Therefore, the probability of not getting 10 heads for one coin is approximately (1-{1 \over 1000}).

This means that the probability of not getting 10 heads for any of 1000 coins is this number multiplied by itself 1000 times, once for every coin. This probability is therefore \approx (1-{1 \over 1000})^{1000}.

This is approximately {1 \over e} since \lim_{n\to\infty} (1-{1\over n})^n = {1\over e}. Numerically, {1 \over e}\approx{1\over 2.718}\approx 0.37.

Therefore, the probability of this not happening, namely that at least one coin of the 1000 coins will give 10 heads, is 1 minus that. This gives us the answer of approximately 0.63 or 63% that I mentioned in the lecture.
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