The Binomial Distribution

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 repeated flips
• 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 σ² = n * p * (1-p)
• The standard deviation of the distribution is σ = √σ²

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) = _nC_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) = _nC_k * p^k * (1-p)^n-k

P(John makes exactly 10 free-throws) = ₁₂C₁₀ * (.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 (σ²)= n * p * (1-p) = 500 * 0.02 * (1-0.02) = 9.8

Standard deviation = √σ² = √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) = ₅C₃ * (.5)³ * (1-.5)^5-3 = 0.3125

P(exactly 4 heads) = ₅C₄ * (.5)⁴ * (1-.5)^5-4 = 0.15625

P(exactly 5 heads) = ₅C₅ * (.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.