S1 – Binomial Distribution Summary
S1 – Binomial Distribution Summary
 A Discrete random variable (DRV) is a discrete set of characteristics
measured/observed in a trial, which may be subject to random variation.
 The sum of the probabilities of each value of the DRV can take is 1.
 The list/table of values of the DRV with their associated probabilities is
called the probability distribution.
Conditions for a Binomial distribution
 Fixed number of independent trials, n
 Only two possible outcomes: success and failure
 Both outcomes have fixed probabilities: P(success) = p and
P(failure) = q, p + q = 1
 The outcome of each trial is independent of the previous trials
 The distribution is denoted B(n, p) where n and p are the parameters of
the distribution
 If the DRV X has a binomial distribution then X  B(n, p)
 The probability or r successes is given by
P(X = r) = nCr pr qn-r
for 0  r  n
 The probabilities for X = 0, 1, 2, …, n are the terms of the binomial
expansion of (p + q)n
 If X  B(n, p), Table 1 “Cumulative binomial distribution function” in the
formula book and Table 1 in the Appendix of the textbook on page 173
give probabilities of P(X  r) for several values of p and n
 The values of p only go up to 0.5 in the tables. To find probabilities for
higher values of p redefine the distribution by interchanging failure and
success.
 If X  B(n, p) then the mean (or expectation) of X is given by μ = np
and the variance is σ2 = np(1 – p) = npq
 A Discrete random variable (DRV) is a discrete set of characteristics
measured/observed in a trial, which may be subject to random variation.
 The sum of the probabilities of each value of the DRV can take is 1.
 The list/table of values of the DRV with their associated probabilities is
called the probability distribution.
Conditions for a Binomial distribution
 Fixed number of independent trials, n
 Only two possible outcomes: success and failure
 Both outcomes have fixed probabilities: P(success) = p and
P(failure) = q, p + q = 1
 The outcome of each trial is independent of the previous trials
 The distribution is denoted B(n, p) where n and p are the parameters of
the distribution
 If the DRV X has a binomial distribution then X  B(n, p)
 The probability or r successes is given by
P(X = r) = nCr pr qn-r
for 0  r  n
 The probabilities for X = 0, 1, 2, …, n are the terms of the binomial
expansion of (p + q)n
 If X  B(n, p), Table 1 “Cumulative binomial distribution function” in the
formula book and Table 1 in the Appendix of the textbook on page 173
give probabilities of P(X  r) for several values of p and n
 The values of p only go up to 0.5 in the tables. To find probabilities for
higher values of p redefine the distribution by interchanging failure and
success.
 If X  B(n, p) then the mean (or expectation) of X is given by μ = np
and the variance is σ2 = np(1 – p) = npq