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Consider the following question:

A coin has the probability of landing of head equal to 1/4 and is flipped 2000 times.

  1. Use the law of large numbers, find a lower bound to the probability that the total number of heads lies between 480 and 520.

As far as I know the law of large numbers simply states that for a large number of iid variables, their summation approaches to the expected value of the variable. Does it state anything about its rate of convergence?

The article on law of large numbers in Wikipedia proves it using Chebyshev's Inequality (one of the proofs). I am not sure whether finding the lower bound using Chebyshev's Inequality would count as "using the law of large numbers".

How to solve this problem?

Community
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Steel
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    For binomial distributions $\mu = np = 500$ and $\sigma = \sqrt{np(1-p)} \approx 20$, so I think it is asking approximately how many results lie within 1 standard deviation of the mean. – JMoravitz Oct 26 '14 at 04:55
  • How do I use "law of large numbers" to find that? – Steel Oct 26 '14 at 14:57
  • With large enough $ n $, it will be very well approximated by a normal distribution with the same mean and standard deviation. – JMoravitz Oct 26 '14 at 16:36
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    Well that is the central limit theorem, isn't it? I wonder if finding the result by that method would classify as "using law of large numbers". Actually, this question has a second part which says "2. Using central limit theorem, find the probability that the total number of heads lies between 480 and 520." – Steel Oct 26 '14 at 21:30
  • I agree with @JMoravitz that you are expected to assume the "law of large numbers" prompts you to use the normal distribution and do a $\pm 1 \sigma$ calculation. The numbers are so nice, it has to be. It is a very reasonable question whether $2000$ is a large number in this case. If it is not, you would expect the tails to be wider than the normal distribution, so your lower bound could fail. – Ross Millikan Oct 27 '14 at 02:43

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I don't think you can do that. I suppose the question intend to test the Chebyshev's inequality
$$\operatorname{P}( \left| \overline{X}_n-\mu \right| \geq \varepsilon) \leq \frac{\sigma^2}{n\varepsilon^2} $$ . However, the problem is $\mu$ in the formula is sample mean, not the real mean.

athos
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