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
$P (X \geq k)$	$1 - P (X \leq k - 1)$
$P (X < k)$	$P (X \leq k - 1)$

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 $n$ .