Page 128 - Probability Demystified

CHAPTER 7 The Binomial Distribution 117

A binomial distribution is obtained from a probability experiment called a
binomial experiment. The experiment must satisfy these conditions:
1. Each trial can have only two outcomes or outcomes that can be
reduced to two outcomes. The outcomes are usually considered as a
success or a failure.
2. There is a fixed number of trials.
3. The outcomes of each trial are independent of each other.
4. The probability of a success must remain the same for each trial.

EXAMPLE: Explain why the probability experiment of tossing three coins is
a binomial experiment.

SOLUTION:
In order to be a binomial experiment, the probability experiment must satisfy
the four conditions explained previously.
1. There are only two outcomes for each trial, head and tail. Depending
on the situation, either heads or tails can be defined as a success and
the other as a failure.
2. There is a fixed number of trials. In this case, there are three trials
since three coins are tossed or one coin is tossed three times.
3. The outcomes are independent since tossing one coin does not
effect the outcome of the other two tosses.
4. The probability of a success (say heads) is 1 and it does not change.
2
Hence the experiment meets the conditions of a binomial experiment.
Now consider rolling a die. Since there are six outcomes, it cannot be
considered a binomial experiment. However, it can be made into a binomial
experiment by considering the outcome of getting five spots (for example) a
success and every other outcome a failure.
In order to determine the probability of a success for a single trial of a
probability experiment, the following formula can be used.

x n x
C ð pÞ ð1 pÞ
n x
where n ¼ the total number of trials
x ¼ the number of successes (1, 2, 3, .. . , n)
p ¼ the probability of a success
The formula has three parts: n C x determines the number of ways a success
x
can occur. ( p) is the probability of getting x successes, and (1 p) n x is the
probability of getting n x failures.