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The binomial distribution describes the probability of obtainingÂ *k*Â successes inÂ *n*Â binomial experiments.

If aÂ random variableÂ *X*Â follows a binomial distribution, then the probability thatÂ *X*Â =Â *k*Â successes can be found by the following formula:

**P(X=k) =Â _{n}C_{k}Â * p^{k}Â * (1-p)^{n-k}**

where:

**n:**Â number of trials**k:Â**number of successes**p:**Â probability of success on a given trialthe number of ways to obtainÂ_{n}C_{k}:Â*k*Â successes inÂ*n*Â trials

The binomial probability distribution tends to be bell-shaped when one or more of the following two conditions occur:

**1.** The sample size (n) is large.

**2.** The probability of success on a given trial (p) is close to 0.5.

However, the binomial probability distribution tends to be skewed when neither of these conditions occur. To illustrate this, consider the following examples:

**Example 1: Sample Size (n) is Large**

The following chart displays the probability distribution for when n =Â **200Â **and p =Â **0.5**.

The x-axis displays the number of successes during 200 trials and the y-axis displays the probability of that number of successes occurring.

Since bothÂ **(1)Â **the sample size is large and **(2)Â **the probability of success on a given trial is close to 0.5, the probability distribution is bell-shaped.

Even when the probability of success on a given trial (p) is not close to 0.5, the probability distribution will still be bell-shaped as long as the sample size (n) is large. To illustrate this, consider the following two scenarios when p = 0.2 and p = 0.8.

Notice how the probability distribution is bell-shaped in both scenarios.

**Example 2: Probability of Success (p) is Close to 0.5**

The following chart displays the probability distribution for when n = **10 **and p =Â **0.4**.

Although the sample size (n = 10) is small, the probability distribution is still bell-shaped because the probability of success on a given trial (p = 0.4) is close to 0.5

**Example 3: Skewed Binomial Distributions**

When neither** (1) **the sample size is large nor **(2)Â **the probability of success on a given trial is close to 0.5, the binomial probability distribution will be skewed to the left or right.

For example, the following plot shows the probability distribution when n =Â **20Â **and p =Â **0.1**.

Notice how the distribution is skewed to the right.

And the following plot shows the probability distribution when n = **20Â **and p =Â **0.9**.

Notice how the distribution is skewed to the left.

**Ending Notes**

Each of the charts in this post were created using the statistical programming language R. Learn how to plot your own binomial probability distributions in R using this tutorial.