• Binomial Distribution
Conditions for Binomial Distribution
• Each trial has two possible outcomes (“success” and
“failure”).
• Each trial has the same probability of a success, denoted by p.
• The n trials are independent.
The binomial random variable X is the nnumber of successes in
the n trials.
Probabilities for a Binomial Distribution
For n independent trials, the probability of x successes equals
P(x) =
n!
p x (1 − p)n−x ,
x!(n − x)!
x = 0, 1, 2, . . . , n
Here p is the probability for a success of each trial.
Guideline for Using Binomial Distribution
For sampling n subjects from a population without replacement, if
n is less than 10% of the population size, then we can use the the
binomial distribution to approximate the exact probability
distribution of the number of successes.
(In other words, we pretend it is sampling WITH replacement.)
Mean and Standard Deviation
The mean µ of the binomial distribution for n trials with “success”
probability p is
µ = np
And the standard deviation is
σ=
p
np(1 − p)
Approximate The Binomial Distribution by The Normal
Distribution
The binomial distribution can be approximated well by the normal
distribution when
(i) n is large enough (≤ 40);
(ii) the expected number of successes np is at least 15; and
(iii) the expected number of failures np is at least 15.