**Binomial Experiment**

A **binomial experiment **is an experiment that has the following properties:

- The experiment consists of
*n*repeated trials - Each trial has only two possible outcomes: success or failure
- The probability of success, denoted
*p*, is the same for each trial - Each trial is independent; the outcome of one trial does not affect the outcome of another

Suppose we flip a coin three times and record the number of times the coin landed on heads. This is an example of a binomial experiment because:

- The experiment consists of 3
- Each flip has only two possible outcomes: heads or tails
- The probability of success (landing on heads) is 0.5 for each flip
- Each flip is independent; the outcome of one flip does not affect the outcome of another

**Binomial Random Variable**

A **binomial random variable** is the number of successes, denoted *x*, in a binomial experiment with *n *repeated trials, and it follows a **binomial distribution**.

For example, if we flip a coin three times, the number of times it lands on heads has the following binomial distribution, where X = number of times it lands on heads and P(X) = probability associated with that outcome:

The binomial distribution has the following properties:

- The mean of the distribution is
**μ**=*n***p* - The variance of the distribution is
**σ**=^{2}*n***p * (1-p)* - The standard deviation of the distribution is
**σ**= √σ^{2}

To find the probability of exactly *x *successes in *n *trials, given the probability of success in a single trial is *p*, we can use the following formula:

P(exactly *x *successes) = _{n}C_{k} * p^{k} * (1-p)^{n-k}

Let’s walk through some examples to gain a better understanding of the binomial distribution.

**Binomial Distribution Example Problems**

**Example 1: John makes 60% of his free-throw attempts. If he shoots 12 free throws, what is the probability that he makes exactly 10?**

**Step 1: Identify the number of trials ( n), the number of successes we’re interested in (k), and the probability of success on each trial (p).**

*n *= number of free-throw attempts = 12

*k *= number of successful free-throws we’re interested in = 10

*p *= probability of successful free-throw on each attempt = 0.6

**Step 2: Plug these numbers into the binomial formula or a binomial calculator**

**Using the formula:**

P(John makes exactly 10 free-throws) = _{n}C_{k} * p^{k} * (1-p)^{n-k}

P(John makes exactly 10 free-throws) = _{12}C_{10} * (.6)^{10} * (1-.6)^{12-10}

P(John makes exactly 10 free-throws) = (66) * (.006) * (.16)

P(John makes exactly 10 free-throws) = **0.063**

**Using the calculator:**

Plug the following numbers into the Binomial Distribution Calculator:

The probability that John makes exactly 10 free-throws is about** 6.3%**.

**Example 2: A factory receives a shipment of nails once per month. On average, 2% of the nails are defective. If there are 500 nails per shipment, what is the mean and standard deviation of the number of defective nails per shipment?**

**Step 1: Use the formulas for the mean and standard deviation of a binomial distribution to answer this question.**

Mean (μ)= *n ** *p = *500 * 0.02 = **10**

Variance (σ^{2})= *n ** *p * (1-p) *= 500 * 0.02 * (1-0.02) = 9.8

Standard deviation = √σ^{2} = √9.8 = **3.13**

The mean number of defective nails per shipment is **10 nails** and the standard deviation is **3.13 nails**.

**Example 3: Ando flips a fair coin 5 times. What is the probability that the coin lands on heads 3 times or more?**

**Step 1: Identify the number of trials ( n), the number of successes we’re interested in (k), and the probability of success on each trial (p).**

*n *= number of flips = 5

*k *= number of times the coin lands on heads = 3

*p *= probability that coin lands on heads in a given flip = 0.5

**Step 2: Since we want to know the probability that the coin lands on heads 3 times or more, we need to find the probability that it lands on heads 3 times, 4 times, and 5 times, then add up these probabilities.**

**Using the formula:**

P(exactly 3 heads) = _{5}C_{3} * (.5)^{3} * (1-.5)^{5-3} = 0.3125

P(exactly 4 heads) = _{5}C_{4} * (.5)^{4} * (1-.5)^{5-4} = 0.15625

P(exactly 5 heads) = _{5}C_{5} * (.5)^{5} * (1-.5)^{5-5} = 0.03125

P(3 heads or more) = P(3 heads) + P(4 heads) + P(5 heads) = **0.5**

**Using the calculator:**

Plug the following numbers into the Binomial Distribution Calculator:

The calculator tells us that the probability of getting 3 *or more *heads (PX ≥ x) is **0.5**, which matches the result we got using the formula.