The word “binomial” literally means “two numbers.” A binomial distribution for a random variable *X* is one in which there are only two possible outcomes, **success** and **failure**, for a finite number of **trials**.

Let’s flesh these concepts out a bit. For example, let’s say you’re a basketball player hoping to make a foul shot. For you, success would be “you make the shot” and failure would be “you miss the shot.” In this example, each foul shot is considered a trial.

Another example involves rolling a standard 6-sided die. Let’s say that we hope to roll a five. In this case, we define success as “rolling a 5” and failure as “not rolling a 5.” In this case, each roll of the die would be a trial.

Note that however we define success and failure, the two events must be **mutually exclusive** and **complementary**; that is, they cannot occur at the same time (mutually exclusive), and the sum of their probabilities is 100% (complementary).

Generally, we define the probability of success as *p*, and the probability of failure as *q*. Because the two events are **/complementary**, *q* = 1 – *p*. In our example of the 6-sided die, our probability of success (rolling a 5) is *p* = ⅙; our probability of failure (not rolling a 5) is *q* = 1 – ⅙ = ⅚.

For the binomial distribution to be applied, each successive trial must be **independent** of the last; that is, the outcome of a previous trial has no bearing on the probabilities of success on subsequent trials. For the roll of a die, we know this to be true: just because a five rolled the last time does not change the probability of rolling a 5 on future rolls; the probability of success remains unchanged at 1 in 6.

Lastly, the binomial distribution is a **discrete** probability distribution. This means that the possible outcomes are distinct and non-overlapping. (For example, when you roll a die, you can roll a 3, and you can roll a 4, but you cannot roll a 3.5. For more on discrete versus continuous distributions, check out this other post on the normal distribution.)

Now that we’ve covered the basic definitions involved for a binomial distribution, let’s briefly summarize them, before looking at an example. A random variable *X* follows a binomial probability distribution if:

1) There are a finite number of trials, *n*.

2) Each trial is independent of the last.

3) There are only two possible outcomes of each trial, success and failure. The probability of success is *p* and the probability of failure is *q*.

4) Success and failure are mutually exclusive (cannot occur at the same time) and complementary (the sum of their probabilities is 100%; *q* = 1 – *p*).

### Example of the binomial distribution using coin flips

Let’s assume we are flipping a coin 6 times. We’ll bet on heads, so success for us is “the coin lands heads” and failure is “the coin lands tails.” In this case, the probability of success and failure are both 0.5:

*p* = 0.5

*q* = 0.5

Now, how could we calculate the probability that the coin comes up heads on 5 out of the 6 trials? There are 6 possible desired outcomes, shown below:

HHHHHT

HHHHTH

HHHTHH

HHTHHH

HTHHHH

THHHHH

Because each trial is independent, the probability of any one of these outcomes occurring is (½)^{6}, or 1/64. Since each outcome is equally probable, and there are six desired outcomes, the probability of 5 out of 6 heads would be 6*(1/64) = 6/64 = 0.09. This is visualized below, in the second bar from the right:

Starting from the left, the other bars show the probability of getting 0 heads, 1 heads, 2 heads, etc., all the way up to 6 heads on the far right. The smooth line represents the normal curve. You can get a sense from this graph that *as the number of trials increases, the binomial distribution will approach the normal distribution*. To illustrate this further, let’s see what happens to the graph when we increase the number of coin flips further, up to 16 and then 160.

We can get an even clearer view here of the binomial distribution approaching the normal distribution as the number of trials, *n*, gets larger and larger. (Note that this will only be the case when the probabilities of success and failure are both equal to 0.5. When p diverges from 0.5, the peak of the distribution will skew either to the left or to the right.)

Now let’s consider the binomial distribution for 100 trials, or 100 coin flips. What if we actually wanted to calculate the probability of flipping a coin 100 times, and getting heads 52 times? You might be thinking to yourself, *calculating that must take forever*! And it would, if we approached it by listing out all the possible desired outcomes with H’s and T’s, like we did previously, when the number of trials was only 6.

Luckily, there is a faster way. How to quickly calculate binomial probabilities for large numbers of trials? We can apply the following formula, where *X* is our random variable, *n* is our number of trials, *k* is our number of successes, and *p* is the probability of success:

From combinatorics, the expression nCr expands to:

The next example will apply this formula, and it will be a bit more technical. The more general reader may feel inclined to skip to the conclusion!

### Applying the formula for the binomial distribution

Let’s go back to our previous question. What is the probability that we flip a coin 100 times, and it lands heads exactly 52 out of the 100 flips? Let’s first define the value of each term in our formula:

- Since we are flipping the coin 100 times, we have 100 independent trials, and
*n*= 100. - Since we defining success to be “the coin lands heads,” and we are calculating the probability of getting 52 heads,
*k*= 52. - As always, the probability of success, or the coin landing heads, is
*p*= 0.5.

Plugging into our formula, we get:

While knowing the formula is useful, there are also all kinds of useful software programs and websites that can perform these calculations for you. If you’re curious, check out this binomial calculator from Vassar Stats. It allows you to plug in different values of n, k, and p, and instantaneously calculates the probabilities for you.

### Conclusion

- The binomial distribution is a discrete probability distribution used when there are only two possible outcomes for a random variable: success and failure.
- Success and failure are mutually exclusive; they cannot occur at the same time.

The binomial distribution assumes a finite number of trials,*n*. - Each trial is independent of the last. This means that the probability of success,
*p*, does not change from trial to trial. - The probability of failure,
*q*, is equal to 1 –*p*; therefore, the probabilities of success and failure are complementary.

Do you need more practice with the binomial distribution? Check out our statistics videos and blog posts here!

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