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# A-Level Maths | Binomial Distribution
- URL: https://www.a-level-maths-tutor.com/a-level-maths-binomial-distribution/
- Published: 2026-03-20T18:58:59.000Z
- Updated: 2026-07-22T11:51:23.000Z
- Author: Jaisul Naik
- Tags: Stats

A binomial distribution models the number of successes in a fixed number of independent trials, where each trial has only two possible outcomes: success or failure.

$X \\sim B(n, p)$ means we are modelling the distribution $X$ binomially, where:

$n$: the number of trials

$p$: the probability of success on each trial

If $X \\sim B(n,p)$, then $P(X=r) = \\dbinom{n}{r}p^r(1-p)^{n-r}$

**Worked Example**

$X \\sim B(10, 0.3)$. Find $P(X=4)$.

$P(X=4) = \\dbinom{10}{4}(0.3)^4(0.7)^6 = 0.2001$

## Question 1

$X \\sim B(20, 0.4)$

**a)** Find $P(X=3)$

**b)** Find $P(X=6)$

**c)** Find $P(X=9)$

**d)** Find $P(X=12)$

**e)** Find $P(X=15)$

Show Answers 

**a)** $0.0123$

**b)** $0.1244$

**c)** $0.1597$

**d)** $0.0355$

**e)** $0.0013$

## Question 2

$X \\sim B(25, 0.35)$

**a)** Find $P(X \\leq 4)$

**b)** Find $P(X \\leq 7)$

**c)** Find $P(X \\leq 9)$

**d)** Find $P(X \\leq 12)$

**e)** Find $P(X \\leq 15)$

Show Answers 

**a)** $0.0320$

**b)** $0.3061$

**c)** $0.6303$

**d)** $0.9396$

**e)** $0.9971$

**Worked Example**

$X \\sim B(18, 0.25)$. Find $P(X < 7)$.

$P(X < 7) = P(X \\leq 6) = 0.8610$

## Question 3

$X \\sim B(30, 0.3)$

**a)** Find $P(X < 5)$

**b)** Find $P(X < 8)$

**c)** Find $P(X < 10)$

**d)** Find $P(X < 13)$

**e)** Find $P(X < 16)$

Show Answers 

**a)** $0.0302$

**b)** $0.2814$

**c)** $0.5888$

**d)** $0.9155$

**e)** $0.9936$

**Worked Example**

$X \\sim B(18, 0.25)$. Find $P(X \\geq 7)$.

$P(X \\geq 7) = 1 - P(X \\leq 6) = 1 - 0.8610 = 0.1390$

## Question 4

$X \\sim B(20, 0.4)$

**a)** Find $P(X \\geq 5)$

**b)** Find $P(X \\geq 8)$

**c)** Find $P(X \\geq 10)$

**d)** Find $P(X \\geq 13)$

**e)** Find $P(X \\geq 16)$

Show Answers 

**a)** $0.9490$

**b)** $0.5841$

**c)** $0.2447$

**d)** $0.0210$

**e)** $0.0003$

## Left tail hypothesis testing

### Question 5

![](https://storage.ghost.io/c/37/0e/370e4698-c6d5-4470-8b24-7d14723fff86/content/images/2026/04/CleanShot-2026-04-14-at-12.11.51@2x.png)

### Question 6

![](https://storage.ghost.io/c/37/0e/370e4698-c6d5-4470-8b24-7d14723fff86/content/images/2026/04/CleanShot-2026-04-14-at-12.12.21@2x-1.png)

### Question 7

![](https://storage.ghost.io/c/37/0e/370e4698-c6d5-4470-8b24-7d14723fff86/content/images/2026/04/CleanShot-2026-04-14-at-12.12.28@2x.png)

### Question 8

![](https://storage.ghost.io/c/37/0e/370e4698-c6d5-4470-8b24-7d14723fff86/content/images/2026/04/CleanShot-2026-04-14-at-12.12.40@2x.png)

## Right tail hypothesis testing

### Question 9

![](https://storage.ghost.io/c/37/0e/370e4698-c6d5-4470-8b24-7d14723fff86/content/images/2026/04/CleanShot-2026-04-14-at-12.13.50@2x.png)

### Question 10

![](https://storage.ghost.io/c/37/0e/370e4698-c6d5-4470-8b24-7d14723fff86/content/images/2026/04/CleanShot-2026-04-14-at-12.13.56@2x.png)

### Question 11

![](https://storage.ghost.io/c/37/0e/370e4698-c6d5-4470-8b24-7d14723fff86/content/images/2026/04/CleanShot-2026-04-14-at-12.14.01@2x.png)

### Question 12

![](https://storage.ghost.io/c/37/0e/370e4698-c6d5-4470-8b24-7d14723fff86/content/images/2026/04/CleanShot-2026-04-14-at-12.14.06@2x.png)

## Two-tail hypothesis testing

### Question 13

![](https://storage.ghost.io/c/37/0e/370e4698-c6d5-4470-8b24-7d14723fff86/content/images/2026/04/CleanShot-2026-04-14-at-12.14.42@2x.png)

### Question 14

![](https://storage.ghost.io/c/37/0e/370e4698-c6d5-4470-8b24-7d14723fff86/content/images/2026/04/CleanShot-2026-04-14-at-12.14.49@2x.png)

## Left tail hypothesis test critical values

### Left tail

### Question 15

![](https://storage.ghost.io/c/37/0e/370e4698-c6d5-4470-8b24-7d14723fff86/content/images/2026/04/CleanShot-2026-04-14-at-13.08.39@2x.png)

## Right tail hypothesis test critical values

### Question 16

![](https://storage.ghost.io/c/37/0e/370e4698-c6d5-4470-8b24-7d14723fff86/content/images/2026/04/CleanShot-2026-04-14-at-13.09.43@2x.png)

## Two-tail hypothesis test critical values

### Question 17

![](https://storage.ghost.io/c/37/0e/370e4698-c6d5-4470-8b24-7d14723fff86/content/images/2026/04/CleanShot-2026-04-14-at-13.09.55@2x.png)

### Question 18

![](https://storage.ghost.io/c/37/0e/370e4698-c6d5-4470-8b24-7d14723fff86/content/images/2026/04/CleanShot-2026-04-14-at-13.10.07@2x.png)

### Question 19

![](https://storage.ghost.io/c/37/0e/370e4698-c6d5-4470-8b24-7d14723fff86/content/images/2026/04/CleanShot-2026-04-14-at-13.10.12@2x.png)

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