Binomial Distribution

Basic Concepts

Suppose an experiment has the following characteristics:

• the experiment consists of n independent trials, each with two mutually exclusive outcomes (success and failure)
• for each trial the probability of success is p (and so the probability of failure is 1 – p)

Each such trial is called a Bernoulli trial. Let x be the discrete random variable whose value is the number of successes in n trials. Then the probability distribution function for x is called the binomial distribution, which is defined as follows:

fx= Cn,xpx1-pn-x

where Cn, x= n!x!n-x! and n factorial is n!=nn-1n-2⋯3∙2∙1.

Cn, x can be calculated in Excel by using the function COMBIN(n,x).

Figure 30 shows a graph of the probability density function for the binomial distribution with n = 10 and p = .25.

30. Binomial distribution

The mean of this distribution is np and the variance is np(1 – p )

Excel Functions

Excel provides the following functions regarding the binomial distribution:

BINOM.DIST(x, n, p, cum) where n = the number of trials, p = the probability of success for each trial and cum takes the value TRUE or FALSE.

BINOM.DIST(x, n, p, FALSE) = probability density function value at x for the binomial distribution, i.e. the probability that there are x successes in n trials where the probability of success on any trial is p.

BINOM.DIST(x, n, p, TRUE) = cumulative probability distribution value at x for the binomial distribution, i.e. the probability that there are at most x successes in n trials where the probability of success on any trial is p.

BINOM.INV(n, p, 1 – α) = the critical value; i.e. the small value of x such that BINOM.DIST(x, n, p, TRUE) ≥ 1 – α.

For versions of Excel prior to Excel 2010, the following functions are used instead: BINOMDIST, which is equivalent to BINOM.DIST, and CRITBINOM, which is equivalent to BINOM.INV.

Excel 2013 introduces the following new function (where x ≤ y ≤ n).

BINOM.DIST.RANGE(n, p, x, y) = the probability there are between x and y successes (inclusive) in n trials where the probability of success on any trial is p.

Thus,

      BINOM.DIST.RANGE(n, p, x, y) = BINOM.DIST(y, n, p, TRUE)

                                                      – BINOM.DIST(x – 1, n, p, TRUE)

and for x > 0

      BINOM.DIST.RANGE(n, p, 0, y) = BINOM.DIST(y, n, p, TRUE)

The y parameter may be omitted, in which case we have

      BINOM.DIST.RANGE(n, p, x) = BINOM.DIST(x, n, p, FALSE).

Example: What is the probability that if you throw a die 10 times it will come up six 4 times?

We can model this problem using the binomial distribution with n = 10 and p = 1/6 as follows

Alternatively the problem can be solved using the Excel formula:

BINOM.DIST(4, 10, 1/6, FALSE) = 0.054266

The probability that the die will come up six at least 4 times can be calculated by the Excel formula

1 – BINOM.DIST(3, 10, 1/6, TRUE) = 0.06728

Hypothesis Testing

Example: Suppose you have a die and you suspect that it is biased towards the number 3, and so you run an experiment in which you throw the die 10 times and count that the number 3 comes up 4 times. Determine whether the die is biased.

The population random variable x = the number of times 3 occurs in 10 trials has a binomial distribution. Let π be the population parameter corresponding to the probability of success on any trial. We define the following null and alternative hypotheses:

H₀: π ≤ 1/6; i.e. the die is not biased towards the number 3

H₁: π > 1/6

Now setting α = .05, we have

P(x ≤ 4) = BINOM.DIST(4, 10, 1/6, TRUE) =  0.984538 > 0.95 = 1 – α.

And so we reject the null hypothesis with 95% level of confidence.

Example: We suspect that a coin is biased towards heads. When we toss the coin 9 times, how many heads need to come up before we are confident that the coin is biased towards heads?

We use the following null hypothesis:

H₀: π ≤ .5

H₁: π > .5

Using a confidence level of 95% (i.e. α = .05), we calculate

BINOM.INV(n, p, 1 – α) = BINOM.INV(9, .5, .95) = 7

And so 7 is the critical value. If 7 or more heads come up then we are 95% confident that the coin is biased towards heads, and so can reject the null hypothesis.

Note that BINOM.DIST(6, 9, .5, TRUE) = .9102 < .95, while BINOM.DIST(7, 9, .5, TRUE) = .9804 ≥ .95.