The binomial distribution

An experiment that satisfies the following four conditions is called a binomial experiment:

1. There are n identical trials. In other words, the given experiment is repeated n times. All these repetitions are performed under identical conditions.

2. Each trial has two and only two outcomes. These outcomes are usually called a success and a failure.

3. The probability of success is denoted by p and that of failure by q,

and . The probabilities p and q remain constant for each trial.

4. The trials are independent. In other words, the outcome of one trial does not affect the outcome of another trial.

It is important to note that one of the two outcomes of a trial is called a success and the other a failure. Note that a success does not mean that the corresponding outcome is considered favorable or desirable. Similarly, a failure does not necessarily refer to an unfavorable or undesirable outcome. Success and failure simply the names used to denote the two possible outcomes of a trial.

The random variable x that represents the number of successes in n trial for a binomial experiment is called a binomial random variable.

Binomial formula:

For a binomial experiment, the probability of exactly x successes in n trials is given by the binomial formula:

where

total number of trials

probability of success

probability of failure

number of success in n trials

number of failures in n trials.

To find the probability of x successes in n trials for a binomial experiment, the only values needed are those of n and q. These are called the parameters of the binomial distribution or simply the binomial parameters.

Example:

A certain drug is effective in 30 per cent of the cases in which it has been prescribed. If a doctor is now administering this drug to four patients, what is the probability that it will be effective for at least three of the patients?

Solution:

We can consider the administration of the drug to each patient as a trial. Thus, this experiment has four trials. There are only two outcomes for each trial: the drug is effective or the drug is not effective. The event

“effective for at least three” can be broken down into two mutually exclusive events (outcomes), “ effective for three or effective for four”.

If we use the term “success” instead of “effective” we can say that

P (at least 3 successes) = P (3 successes or 4 successes)=

=P (3 successes)+ P (4 successes)= P (x =3)+ P (x =4).

Now we can find and separately. Since the drug is effective in 30% of the cases, we say that the probability that the drug is effective in any single case is p =0.3.

Hence, , then the equation of the particular binomial distribution is

Hence, we have

.

Practically interpreted, this number means that if the drug is administrated to 10 000 sets of four patients, in about 837 of the 10 000 sets will be effective for at least three patients out of four.

Example:

It is known from past data that despite all efforts, 2% of the packages mailed through post office do not arrive at their destinations within the specified time. A corporation mailed 10 packages through post office.

a) Find the probability that exactly one of these 10 packages will not arrive at this destination within the specified time.

b) Find the probability that at most one of these 10 packages will not arrive at this destination within the specified time.

Solution:

Let us call it a success if a package does not arrive at its destination within the specified time and a failure if it does arrive within the specified time. Then

n =10; p= 0.02; q =0.98

a) For this part,

x = number of successes=1

n - x = number of failures=10-1=9

Substituting all values in the binomial formula, we obtain:

.

Thus, there is a 0.1667 probability that exactly one of the 10 packages mailed will not arrive at its destination within the specified time.

b) The probability that at most one of the 10 packages will not arrive at its destination within the specified time is given by the sum of the probabilities of x =0 and x =1. Thus,

.

Thus, the probability that at most one of the 10 packages will not arrive at its destination within the specified time is 0.9838.

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