- Bernoulli trial: a random variable such that;
- Two possible outcomes, success and failure;
- Probability of success is always the same (P(success) = p);
- All trials are independent.
Examples:
- fair coin (50% chance of heads and tails), heads = success, p = 1/2
- roll a fair die, success = 4, p = 1/6
- have 1m voters, 70% favor one, pick one voter, success = he favors Christie, p = 7/10
If you repeat a Bernoulli trial n times and let X be the number of success in the trial, then X is a binomial random variable.
If you flip a coin twice and say that X = # of heads, then X is a binomial random variable
X = number of successes, random
| X | P(X) | X * P(X) | (X - E(X))^2 * P(X) |
|---|---|---|---|
| 0 | 1 - p | 0 | (0 - 1)^2 * (1 - P) |
| 1 | p | p | (1 - P)^2 - P |
Std. Dev = sqrt(p - p^2)
If I perform a B trial with prob of successes = p n times P(x = x) = (n! / (x!(n-x)!)) * p^x * (1 - p)^(n - x)
E.g. probability of rolling die 5 times, getting 4 two times
binompdf(n = 5, p = 1/6, x = 2)
binomcdf prob less than or equal to x = 2