BINOMIAL DISTRIBUTIONS
(Bernoulli Trials)
Example #1
A die is tossed 5 times. Determine the probability that
the number 3 is rolled twice.
BERNOULLI TRIALS: repeated independent trials measured
in terms of successes or failures
THREE CRITERIA of a BERNOULLI TRIAL:
1.
2.
3.
Exactly two outcomes – success or failure.
Each trial is independent.
The probability of each outcome is the same for each trial.
Bernoulli trials follow a binomial distribution. The random variable is
the number of successes in a given number of trials.
The probability of x successes in n trials is
 n  x n x
P(x) =   p q
x
where p is the probability of success for each trial
q is the probability of failure (1 – p)
The expectation (expected value) for a binomial distribution is
E(x) = np
where n is the number of trials
p is the probability of success
Example #2 Consider a binomial distribution with p = 0.4 and n = 3.
a)
Complete the given table.
x
0
b)
Determine the expected
value for the distribution.
P(x)
1
2
3
Example #3 On a quiz of 7 multiple choice questions, a student has no idea
of the correct answer and must guess. If there are 5 choices
for each question:
a) Determine the probability of a success or a failure.
b) Determine the probability that a student gets:
i) 5 answers correct
ii) 3 answers incorrect
c) Determine the probability that a student gets at least 2 answers correct.
d) Determine the expected number of correct answers.