A special and useful discrete probability distribution is the binomial distribution. Before introducing binomial distribution, we first introduce Bernoulli trials.

Chance experiments satisfying the following conditions are called Bernoulli trials:

1. Only two possible outcomes: success and failure.
2. The probability of success, p, is the same on every trial.
3. Trials are independent. Observing a head on the current trial won’t affect the result of the next trial. If trials are not strictly independent, it is still okay as long as the sample is less than 10% of the population.
4. The number of trials is fixed.

4.5.1 Probability Distribution of a Binomial Random Variable

Let [latex]X[/latex] be the number of successes in a sequence of n Bernoulli trials with probability of success p. Then [latex]X[/latex] follows a binomial distribution with the probability distribution given by

[latex]P(X=x) =_nC_x p^x (1-p)^{n-x}, \quad x= 0, 1, ... , n.[/latex]

Note that even though the probability distribution is not given in the table form, it lists all possible values of x and their probabilities.

Interpretation of the binomial probability equation is as follows: There are [latex]n[/latex] Bernoulli trials and hence [latex]n[/latex] outcomes that are either a success or a failure. We want to find the probability of observing [latex]x[/latex] successes. If we observe [latex]n[/latex] successes, there must be [latex](n-x)[/latex] failures. The probability of success is [latex]p[/latex] and the probability of failure is [latex](1-p)[/latex]. Hence, the probability of [latex]x[/latex] consecutive successes followed by [latex](n-x)[/latex] failures is [latex]p^x(1-p)^{n-x}[/latex] because the trials are independent and thus the special multiplication rule applies. Next, observe that there are [latex]_nC_x[/latex] ways to rearrange [latex]x[/latex] successes and [latex](n-x)[/latex] failures and so the probability of [latex]x[/latex] successes, disregarding order, is [latex]_nC_xp^x(1-p)^{n-x}[/latex]. The term [latex]_nC_x[/latex] is called the binomial coefficient. For example, if we conduct the trial three times and observe one success, it can occur in the first, second, or third trials. The number of ways to observe one success out of three trials is [latex]_3C_1=3[/latex].

4.5.2 Mean and Standard Deviation of Binomial Distribution

Recall that the general formulas for the mean and the standard deviation of a discrete variable are:

[latex]\mu=\sum xP(X=x), \quad \sigma=\sqrt{\sum x^2P(X=x)-\mu^2}.[/latex]

And the probability distribution of a binomial random variable is given by

[latex]P(X=x)=_nC_x p^x(1-p)^{n-x}, x=0, 1, \cdots, n.[/latex]

Plugging the third equation in the first and second equations gives the mean and standard deviation of a binomial random variable:

[latex]\begin{align*} \mu&=\sum xP(X=x)=\sum x [\underbrace{_nC_x \times (p^x)\times (1-p)^{n-x}]}\limits_{\mbox{prob dist'n of binomial}}=np;\\ \sigma&=\sqrt{\sum x^2P(X=x)-\mu^2}=\sqrt{\sum x^2 \underbrace{[(_nC_x) \times (p^x)\times (1-p)^{n-x}]}\limits_{\mbox{prob dist'n of binomial}}-(np)^2}=\sqrt{np(1-p)}. \end{align*}[/latex]

Note that we can use  [latex]\mu = np, \sigma = \sqrt{ np(1-p) }[/latex] to find the mean and standard deviation of a discrete random variable [latex]X[/latex] if and only if [latex]X[/latex] follows a binomial distribution with parameters [latex]n[/latex] and [latex]p[/latex].

4.5.3 Steps to Find Probabilities Related to Binomial Distribution

In order to apply the binomial probability formula, we need to make sure that the variable follows a binomial distribution by checking:

1. Does each trial in the experiment have only two possible outcomes?
2. Are the trials independent?
3. Does each trial have the same probability of success?
4. Is the number of trials fixed?

If the answers to all the above questions are yes and we perform n trials, let [latex]X=[/latex] # of successes, then we can claim that [latex]X[/latex] follows a binomial distribution. We can apply the binomial probability formula as follows:

1. Identify the success event.
2. Determine the probability of success p.
3. Determine n the total number of trials.
4. Write down the event of interest in terms of the binomial variable [latex]X[/latex].
5. Apply the binomial probability formula [latex]P(X=x) = _nC_xp^x(1-p)^{n-x}[/latex] to calculate the probability of each outcome in the event, and then add these probabilities together.