Suppose we toss a coin 100 times Which is bigger the probabi

Suppose we toss a coin 100 times. Which is bigger, the probability of exactly 50 heads or at least 60 heads?

Solution

As you get more and more samples, you can use the Normal Distribution instead of the Binomial Distribution to figure out probabilities. Basically you look at the normal curve (bell shaped curve) and try to figure out the area under the curve you are looking for. In this case the bell shaped curve has it\'s peak at 50, and starts at 0 and ends at 100. We are looking to find the area under the curve from 60 and above. We need to figure out how many standard deviations away we are from 50. It turns out the formula for the standard deviation is sqrt(n * p * q) where n is the number of trials, p is the probability of success and q is the probability of failure. In this case p = 1/2, n = 100, q = 1/2, so the standard deviation size is sqrt(100 * 1/2 * 1/2) = 5. 60 is two standard deviations from 50.

The general normal distribution says that the probability of being within two standard deviations is approximately 95.45%. This corresponds to being between 40 and 60 heads. The probability of this not happening is 1 - 95.45%, or 4.55%. That number includes being both below 40 heads and more than 60 heads. The binomial distribution is symmetric when p = 1/2, so cut the 4.55% in half to get approximately 2.77% probability. As to whether or not it is 60 and greater or just \"greater\", it is an approximation, and the exact value of 60 will probably not change the value by more than 1/100 of a percent (a guess).

so prob of exact 50 is more