1. Bernoulli Trials Two Possible Outcomes Trials are independent.
Success, with probability p
Failure, with probability q = 1  p
Trials are independent.

2. Binomial Distribution
For n Bernoulli trials, the number of successes X is a binomial random variable. The probability of k successes is given by the binomial probability formula:
As k varies with fixed n and p, the binomial probabilities define a binomial probability distribution over {0, 1, 2, …, n}.

4. Mean and Standard Deviation of a Binomial RV

5. Law of Large Numbers
Informal: If n is large, the proportion of successes in n Bernoulli trials will be very close to p.
Formal: For Bernoulli trials with n and p, as n  , for all  > 0, where k is the number of successes in the n trials.