1 A fair coin is tossed 100 times. The expected number of heads is 50, and the standard deviation for the number of heads is (100·1/2·1/2)1/2 = 5. What does Chebyshev’s Inequality tell you about the probability that the number of heads that turn up deviates from the expected number 50 by three or more standard deviations (i.e., by at least 15)?

Using Chebyshev’s Inequality,

P(|X−μ|≥3σ)

=P(|X−50|≥15)≤ 5^2 / 15^2

=25/225

=1/9

2 Write a program that uses the function binomial(n,p,x) to compute the exact probability that you estimated in Exercise 1. Compare the two results.

pbinom(50, 100, 0.5)
## [1] 0.5397946

The results from the two is very different.