# Binomial probability distribution

> IB Mathematics AA SL · Statistics & Probability
> Source: https://www.owlsprep.com/study/ib-math-aa-sl-u4-binomial-probability-distribution/

This subtopic introduces the discrete binomial probability distribution, one of the most widely used distributions for independent repeated trials. You will learn to identify valid binomial settings, calculate probabilities, and find key distribution parameters.

**Prerequisites:** [Combinations and binomial coefficients](https://www.owlsprep.com/study/ib-math-aa-sl-u3-combinations/); [Discrete random variables](https://www.owlsprep.com/study/ib-math-aa-sl-u4-discrete-random-variables/); Basic probability rules

## Learning objectives

- Identify valid binomial distribution settings using the core conditions
- Calculate binomial probabilities (individual and cumulative) using formula and GDC
- Find the expected value, variance and standard deviation of a binomial distribution
- Solve real-world applied problems involving binomial probability

## Conditions for a Binomial Distribution

**Binomial Random Variable** — A discrete random variable that counts the number of successful outcomes in a fixed sequence of independent identical trials of a random experiment.

*Notation:* $X \sim \text{B}(n, p)$ where $n$ = number of trials, $p$ = probability of success per trial

> **BINS Mnemonic for Binomial Conditions**
>
> All four conditions must be met for a distribution to be binomial:
> 1. **B**inary: Each trial only has two outcomes (success/failure)
> 2. **I**ndependent: Trials do not affect each other
> 3. **N**umber fixed: You have a set, bounded number of trials $n$
> 4. **S**ame probability: The probability of success $p$ is constant across all trials

**Worked example:** Determine whether the scenario below is binomial: Flipping a biased coin with $P(\text{heads}) = 0.6$ 15 times, counting the number of heads obtained. If binomial, state $n$ and $p$.

1. Check each BINS condition one by one:
2. 1. Binary: Each flip has two outcomes: heads (success) or tails (failure). Condition met.
3. 2. Independent: The result of one flip does not change the probability of other flips. Condition met.
4. 3. Fixed number of trials: $n = 15$, fixed. Condition met.
5. 4. Constant probability: $p = 0.6$ for all flips. Condition met.
6. Conclusion: This is a binomial distribution with parameters:
7. $$n = 15, \quad p = 0.6$$

**Check your understanding**

Which of these scenarios is binomial?

1. Rolling a fair die 10 times, counting the number of sixes rolled

   - Yes, it is binomial
   - No, it is not binomial

   *Answer:* Yes, it is binomial

   *Why:* Correct: All BINS conditions are satisfied.

2. Surveying 50 people to find how many hours they exercise per week

   - Yes, it is binomial
   - No, it is not binomial

   *Answer:* No, it is not binomial

   *Why:* Correct: Outcomes are not binary, so this is not binomial.

## Binomial Probability Formula

**Binomial Probability Mass Function** — The probability of getting exactly $k$ successes in $n$ independent trials is given by the formula below.

*Example:* Used to calculate $P(X=k)$ for exact values of $k$

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

$$\binom{n}{k} = \frac{n!}{k!(n-k)!}$$

**Worked example:** If $X \sim \text{B}(4, 0.25)$, find $P(X=2)$.

1. Identify parameters: $n=4$, $k=2$, $p=0.25$, $1-p=0.75$
2. Calculate the binomial coefficient:
3. $$\binom{4}{2} = \frac{4!}{2!2!} = 6$$
4. Substitute into the binomial probability formula:
5. $$P(X=2) = 6 \times (0.25)^2 \times (0.75)^2$$
6. Simplify to 3 significant figures:
7. $$P(X=2) = 0.211$$

> **tip**
>
> In IB exams, you can use your GDC to calculate binomial probabilities directly. Always verify your manual calculation with the GDC to avoid arithmetic errors.

## Expected Value and Variance

For any binomial distribution, we can use simple derived formulas to find the expected value (mean) and variance, without needing to sum over all possible values of $X$.

**Binomial Distribution Parameters** — For $X \sim \text{B}(n,p)$, the expected value (mean) and variance are given by these simple formulas:

- Expected value: $E(X) = np$
- Variance: $\text{Var}(X) = np(1-p)$
- Standard deviation: $\sigma = \sqrt{np(1-p)}$

**Worked example:** A basketball player has a 0.8 probability of making a free throw, independent of previous attempts. If she takes 10 free throws, find the expected number of baskets and the standard deviation.

1. Define the random variable: Let $X$ = number of made baskets. $X \sim \text{B}(10, 0.8)$
2. Calculate the expected number of baskets:
3. $$E(X) = np = 10 \times 0.8 = 8$$
4. Calculate the variance first:
5. $$\text{Var}(X) = np(1-p) = 10 \times 0.8 \times 0.2 = 1.6$$
6. Find the standard deviation, rounded to 3 significant figures:
7. $$\sigma = \sqrt{1.6} \approx 1.26$$

## Cumulative Binomial Probability Problems

Most IB exam questions ask for cumulative probabilities, such as "at most 3 successes" or "at least 5 successes", rather than the probability of exactly $k$ successes. You can calculate these by summing individual probabilities, or directly using your GDC's cumulative binomial function.

> **exam_tip**
>
> Always rewrite the inequality from the question explicitly to avoid off-by-one errors. For example: $P(X \geq k) = 1 - P(X \leq k-1)$, and $P(X < k) = P(X \leq k-1)$.

**Worked example:** 70% of voters in a population support a new policy. If we select a random sample of 12 voters, find the probability that at least 8 voters support the policy, rounded to 3 significant figures.

1. Define the random variable: Let $X$ = number of supporters, $X \sim \text{B}(12, 0.7)$
2. Rewrite "at least 8" as an inequality: $X \geq 8$. Rearrange for cumulative probability:
3. $$P(X \geq 8) = 1 - P(X \leq 7)$$
4. Use GDC to find the cumulative probability $P(X \leq 7) \approx 0.3487$
5. Calculate the final result:
6. $$P(X \geq 8) = 1 - 0.3487 \approx 0.651$$

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Assuming any count of successes is binomial when sampling without replacement from a small population
  - Why it fails: Trials are not independent because removing an item changes the probability for the next trial
  - Correct: Use the hypergeometric distribution for small populations, or approximate as binomial only if your sample is <10% of the total population
- **Wrong:** Calculating $P(X \geq k) = 1 - P(X \leq k)$ for "at least k successes"
  - Why it fails: This is an off-by-one error that includes or excludes the wrong bound
  - Correct: Always explicitly rewrite the inequality: $P(X \geq k) = 1 - P(X \leq k-1)$
- **Wrong:** Swapping $p$ and $1-p$ when calculating expected value
  - Why it fails: Confusing the probability of success and failure for the outcome you are counting
  - Correct: $E(X) = n \times p$, where $p$ is the probability of the outcome you are counting, regardless of whether it is labelled "success"
- **Wrong:** Treating a binomial random variable as continuous
  - Why it fails: Confusing binomial with the normal distribution used for approximations
  - Correct: Binomial distribution is always discrete, as it counts whole numbers of successes

## Cheatsheet

| Property | Rule/Formula |
| --- | --- |
| Notation | $X \sim \text{B}(n, p)$ |
| Conditions | B: Binary, I: Independent, N: Fixed n, S: Constant p |
| $P(X=k)$ | $\binom{n}{k} p^k (1-p)^{n-k}$ |
| Expected value | $E(X) = np$ |
| Variance | $\text{Var}(X) = np(1-p)$ |
| $P(X \leq k)$ | Use GDC cumulative binomial function |
| $P(X \geq k)$ | $1 - P(X \leq k-1)$ |
| $P(X < k)$ | $P(X \leq k-1)$ |

## What's next

Binomial probability is a core foundation for almost all applied statistical analysis, from opinion polling to quality control in manufacturing. Mastering the binomial conditions also helps you recognize when a scenario does not fit a binomial model. Binomial distributions are also the basis for inference for proportions, a key topic in introductory university statistics and hypothesis testing. This topic also leads naturally into the normal distribution, which we use to approximate binomial probabilities for large values of $n$.

- [Normal probability distribution](https://www.owlsprep.com/study/ib-math-aa-sl-u4-normal-probability-distribution/)
- [Calculus](https://www.owlsprep.com/study/ib-math-aa-sl-u5-overview/)
- [Introduction to limits and differentiation](https://www.owlsprep.com/study/ib-math-aa-sl-u5-introduction-to-limits-and-differentiation/)

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