Suppose that a fair coin is tossed 20 times. (The word ‘fair’ here means that each side of the coin has the equal probability of appearing.) If it turned out that the coin came up heads 18 times (out of 20 times of tossing), we would possibly be surprised. Why? Although we know that it can happen by chance, it happens very rarely. It is unlikely, but not totally impossible. If the coin is perfectly fair, we’ll expect that the frequency of occurrence of each side is around 10. Statistically speaking, the probability that the coin comes up heads 18 times is very small. Furthermore, the probability of heads appearing 18 times is much smaller than the probability of heads appearing 10 times. If the coin is fair, it is more expectable that heads appear 10 times than they appear 18 times. Suppose that out of the 20 tosses, the coin comes up heads 13 times. Still we might not be as surprised by the results as we would be if 18 heads had appeared. However, if the coin is fair, then the probability that 10 heads appear is greater than the probability that 13 heads appear. In fact, if a coin is tossed 20 times, the frequency with which heads appear ranges from 0 to 20. What differs is the probability that each frequency appears: the probability of 10 heads appearing is not equal to the probability of 13 heads appearing..

So, what is the probability that the coin comes up heads 10 times? What is the probability that 13 heads appear? Such questions are related to the theoretical probability distribution called binomial distribution.

Binomial distribution is always associated with the experiment called Bernoulli Process. To examine the distribution further, let’s begin with what some understanding of the Bernoulli Process. A Bernoulli process possesses the following properties.

1. The experiment consists of n repeated trials. [Each trial is called a Bernoulli trial.]

2. Each trial results in an outcome that may be classified as a success or a failure.

3. The probability of success, denoted by p, remains constant from trial to trial.

4. The repeated trials are independent.

Let’s look at an example of a Bernoulli process. Suppose that there are 2 blue and 3 black balls in a bag. The experiment is carried out by selecting a ball from the collection of balls 7 times at random, one ball at each time. Therefore, there are 7 repeated trials, each trial is performed by selecting a ball at random from the collection. Before the second trial is performed, the ball selected previously is put back into the bag. Similarly, before the third trial is performed, the ball selected at the second trial is put back into the bag, and so on. (We call this *sampling with replacement*.) Each time we take a ball out of the bag, there are only 2 possible outcomes: either we get a blue or a black. We may consider “getting a black” as a success and “getting a blue” a failure. In other words, we name “getting a black” a success and “getting a blue” a failure. (Of course, we may exchange the naming.) Therefore, the second property of the process is met. The probability that a success occurs is . p is constant since the sampling is conducted with replacement. Just before each trial, there are always 3 black balls among the total of 5 balls. The third property is met. Finally, as you may verify, the repeated trials are independent. The probability that a success occurs in the second trial does not depend on whether a success occurs in the first trial or not. The probability that a success occurs in the third trial does not depend on whether successes occur in the previous trials or not, and so on. The fourth property is satisfied.

How is the binomial distribution is related to Bernoulli process? The probability that x successes occur in a Bernoulli process is B(x), where . B(x) is referred to as probability function of binomial distribution. Here means the number of combinations of n distinct items taken r at a time.

**Sample Problem #1**

A fair coin is tossed 20 times. What is the probability that there will appear 10 heads?

Answer

In this case, n = 20. If we name “a Head (H) appears” a success, what we are seeking is the probability that 10 successes occur, i.e. B(10). Since the coin is fair, the probability that a success occurs is 0.5 in each trial. Therefore, p = 0.5. Applying the formula , we have:

.

**Sample Problem #2**

In Sample Problem #1, find the probability that there will appear 18 heads.

Answer

As in Sample Problem #1, n = 20 and p = 0.5. But now x = 18 and what to calculate is B(18). Applying the same formula, we have:

.

[These two sample problems verify the explanation at the beginning part of this post. More specifically, the probability that the coin comes up 18 heads is very small. It is much smaller than the probability that the coin comes up heads 10 times.]