Binomial probability distribution

IB Mathematics AA SL· 6 min read

1. Conditions for a Binomial Distribution★★☆☆☆⏱ 20 min

📘 Definition

Binomial Random Variable

$X \sim B (n, p)$ where $n$ = number of trials, $p$ = probability of success per trial

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

📐 Worked Example

Determine whether the scenario below is binomial: Flipping a biased coin with $P (heads) = 0.6$ 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
1. Independent: The result of one flip does not change the probability of other flips. Condition met.
4
1. Fixed number of trials: $n = 15$ , fixed. Condition met.
5
1. Constant probability: $p = 0.6$ for all flips. Condition met.
6
Conclusion: This is a binomial distribution with parameters:
7
$n = 15, \quad p = 0.6$

✓ Quick check

Which of these scenarios is binomial?

Rolling a fair die 10 times, counting the number of sixes rolled
- Yes, it is binomial
- No, it is not binomial
Surveying 50 people to find how many hours they exercise per week
- Yes, it is binomial
- No, it is not binomial
Reveal answer
1 —
Correct: Outcomes are not binary, so this is not binomial.

2. Binomial Probability Formula★★★☆☆⏱ 25 min

📘 Definition

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 $X \sim B (4, 0.25)$ , 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
$P(X=2) = 0.211$

3. Expected Value and Variance★★☆☆☆⏱ 20 min

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 $X$ .

📘 Definition

Binomial Distribution Parameters

For $X \sim B (n, p)$ , the expected value (mean) and variance are given by these simple formulas:

Expected value: $E (X) = n p$
Variance: $Var (X) = n p (1 - p)$
Standard deviation: $σ = n p (1 - p)$

📐 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. $X \sim B (10, 0.8)$
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
$\sigma = \sqrt{1.6} \approx 1.26$

4. Cumulative Binomial Probability Problems★★★★☆⏱ 25 min

✓ Calculator OK

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.

📐 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, $X \sim B (12, 0.7)$
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
$P(X \geq 8) = 1 - 0.3487 \approx 0.651$

5. Common Pitfalls

Wrong move:

Assuming any count of successes is binomial when sampling without replacement from a small population

Why:

Trials are not independent because removing an item changes the probability for the next trial

Correct move:

Use the hypergeometric distribution for small populations, or approximate as binomial only if your sample is <10% of the total population

Wrong move:

Calculating $P (X \geq k) = 1 - P (X \leq k)$ for "at least k successes"

Why:

This is an off-by-one error that includes or excludes the wrong bound

Correct move:

Always explicitly rewrite the inequality: $P (X \geq k) = 1 - P (X \leq k - 1)$

Wrong move:

Swapping $p$ and $1 - p$ when calculating expected value

Why:

Confusing the probability of success and failure for the outcome you are counting

Correct move:

$E (X) = n \times p$ , where $p$ is the probability of the outcome you are counting, regardless of whether it is labelled "success"

Wrong move:

Treating a binomial random variable as continuous

Why:

Confusing binomial with the normal distribution used for approximations

Correct move:

Binomial distribution is always discrete, as it counts whole numbers of successes

6. Quick Reference Cheatsheet

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

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2022 · 1
Calculate binomial probability
2021 · 2
Expected value and cumulative probability
2023 · 1
Identify binomial setting, find probability

What's Next

Binomial probability is a core foundation for almost all applied statistical analysis, from opinion polling to quality control in manufacturing. Mastering binomial conditions prepares you to recognize when other distributions like the Poisson or hypergeometric are more appropriate for a problem. 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 $n$ .