Topic 4.10 Notes – Introduction to the Binomial Distribution

1. What a Binomial Random Variable Is

A binomial random variable, $X$, counts the number of successes in a fixed number of trials.

Think: “How many successes out of $n$ tries?”

For a setting to be binomial, all four BINS conditions must hold.

Binary outcomes

Each trial has only two possible outcomes.
You must define what counts as success in context.

• Example: In a clinical trial, “success” might mean a patient’s symptoms improve.
• If there are more than two outcomes, you must group them into success vs. failure.

Independent trials

One trial does not affect another.

• If sampling without replacement, check the 10% condition:
  $n \le 0.10N$ so independence is reasonable.
• On FRQs, if independence is questionable and you don’t address it, you lose credit.

Number of trials fixed

The value of $n$ is set before the experiment begins.

• If you keep going until you get 5 successes, that’s not binomial.

Same probability $p$

The probability of success stays constant.

• Probability of failure is $1 - p$.

Parameters and Notation

• $n$ = number of trials
• $p$ = probability of success
• $X$ = number of successes
• Possible values of $X$: $0,1,2,\dots,n$

If even one BINS condition fails, it’s not binomial. That’s a very common multiple-choice trap.

2. The Binomial Probability Formula

When the setting is binomial, the probability of exactly $x$ successes is:

$$P(X = x) = \binom{n}{x} p^x (1 - p)^{n - x}$$

Let’s break it apart:

• $\binom{n}{x}$ counts how many different ways you can arrange $x$ successes among $n$ trials.
• $p^x$ is the probability of those $x$ successes.
• $(1-p)^{n-x}$ is the probability of the remaining failures.

Multiply them together and you get the probability of exactly $x$ successes.

Quick Example

Suppose a basketball player makes 70% of free throws. She takes 6 shots.
Let $X$ = number of made shots.

Find $P(X = 4)$:

$$P(X=4)=\binom{6}{4}(0.7)^4(0.3)^2$$

$$= 15(0.2401)(0.09) \approx 0.324$$

In context: The probability she makes exactly 4 out of 6 free throws is about 0.324.

Notice how we always state probability in context. That habit matters on the AP exam.

3. Finding Binomial Probabilities

Most of the time you’ll use your calculator.

Exactly x

Use:
binompdf(n, p, x)

“pdf” → single value only.

At most, at least, more than, between

Translate the wording carefully.

• At most 3 → $P(X \le 3)$ → binomcdf(n,p,3)
• Between 2 and 5 → $P(2 \le X \le 5)$
  = $P(X \le 5) - P(X \le 1)$

Calculators only accumulate from 0 up to a value. That’s why complements are helpful.

For example:

$$P(X \ge 4) = 1 - P(X \le 3)$$

Students often forget the “minus 1” when converting “at least.”
“At least 4” means subtract through 3.

If you’re ever unsure, write the full list of possible values and circle what you want.

4. Estimating Binomial Probabilities with Simulation

You can build a probability distribution two ways:

• Theoretical → using the formula
• Empirical → using simulation

Simulation is especially useful when modeling randomness or checking reasoning.

Here’s how you might simulate a binomial situation:

Suppose $p = 0.4$ and $n = 5$.

Step-by-step simulation

1. Assign digits to outcomes.
   • 0-3 = success (40%)
   • 4-9 = failure (60%)
2. Generate 5 random digits → one trial set.
3. Count successes. That’s one value of $X$.
4. Repeat many times.
5. Estimate probability:

$$\frac{\text{number of times event occurs}}{\text{total repetitions}}$$

More repetitions → estimates get closer to the true binomial probability.

This connects to the bigger idea in Unit 4: probabilistic reasoning lets us anticipate patterns in random data. Simulation shows that pattern emerging.

On free-response questions, you must clearly describe:

• How digits were assigned
• How many trials per repetition
• How results were recorded

Vague simulation descriptions don’t earn full credit.