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)?
Chebyshev’s Inequality formula : \(P(|X-\mu|\ge k*\sigma)\le (1/k^2)\)
\(\mu=50\) \(\sigma=5\)
In this case, using the example at least 15 times. \(P(|X-50)\ge 15)\) is the probability we are looking for.
m <- 50
sd <- 5
k <- 15/sd
k## [1] 3The upper bound of the probability;
p <- (1/k^2)
p## [1] 0.1111111The probability of number of heads that turn up deviates from 50 by at least 15 standard deviation is 0.11