Bernoulli and binomial distributions - Mathematics (9660) - Oxford AQA A-Level

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Introduction: Mastering Bernoulli and Binomial Distributions

Welcome to one of the most practical and frequently tested areas of Statistics! This chapter is all about modelling experiments or scenarios where there are only two possible outcomes: success or failure. Think of flipping a coin, checking if a product is faulty, or answering a True/False question.

The concepts of Bernoulli and Binomial distributions are essential tools that allow us to calculate the exact probability of achieving a specific number of successes in a set number of attempts. Don't worry if probability felt abstract before—these distributions give us concrete frameworks to analyze real-world randomness!

1. The Bernoulli Trial: The Simplest Event

1.1 What is a Bernoulli Trial?

The Bernoulli distribution is the foundation of the Binomial distribution. It models a single, simple experiment that meets the following strict condition:

There are exactly two mutually exclusive outcomes:
- Success (usually denoted $X=1$)
- Failure (usually denoted $X=0$)

The probability of success is denoted by $p$. Consequently, the probability of failure is $1-p$, which we often call $q$.

Example: If you roll a standard die and define "Success" as rolling a 6, then $p = 1/6$. "Failure" is rolling anything else, so $q = 5/6$.

1.2 Mean and Variance of a Bernoulli Trial

Since the Bernoulli distribution only involves one trial, its expected value (mean) and variance are simple. You need to know these definitions and how they are derived (though formal derivation may not be required in the exam, understanding the logic helps!).

Key Formulae (Bernoulli Distribution):

Mean (Expected Value):
$$E(X) = p$$

Variance:
$$Var(X) = p(1 - p) = pq$$

Quick Review: The mean of a single Bernoulli trial is simply the probability of success, $p$. If $p=0.7$, then on average, the result of the trial is 0.7 (closer to 1/Success than 0/Failure).

2. The Binomial Distribution: Repetitive Successes

The Binomial distribution models the number of successes, $X$, when you perform a fixed number of independent Bernoulli trials.

Analogy: If Bernoulli is a single coin flip, Binomial is flipping the coin 10 times and counting how many heads you get.

2.1 The Four Conditions for a Binomial Distribution

Before you can use the Binomial distribution $X \sim B(n, p)$, your scenario must satisfy the following four crucial conditions. If any condition fails, you cannot use this model!

Memory Aid: "BINS"

Binary Outcomes: Each trial must have only two outcomes (Success/Failure).
Independence: The outcome of one trial must not affect the outcome of any other trial.
Number of trials (Fixed): There must be a fixed, predetermined number of trials, $n$.
Success Probability ($p$): The probability of success, $p$, must be the same for every trial.

Did you know?

If the trials are not independent (e.g., drawing cards without replacement), you would use the Hypergeometric Distribution instead (though this is outside the scope of AS/A Level 9660).

2.2 Binomial Notation and Parameters

A random variable $X$ that follows a Binomial distribution is written as:

$$X \sim B(n, p)$$

Where:

$n$ is the number of trials (or attempts).
$p$ is the probability of success in a single trial.
$X$ is the specific number of successes we are interested in, where $x = 0, 1, 2, \dots, n$.

Example: A student takes a 10-question multiple-choice quiz. Each question has a 20% chance of being guessed correctly. Let X be the number of correct answers. Then $X \sim B(10, 0.2)$.

3. Calculating Binomial Probabilities

3.1 The Binomial Probability Formula

To find the probability of getting exactly $x$ successes in $n$ trials, we combine two components: the probability of that specific sequence occurring, and the number of ways that sequence can occur.

The probability mass function (PMF) is:

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

Let's break down what each part means:

$\binom{n}{x}$ (pronounced "n choose x"): This is the number of ways to arrange exactly $x$ successes among $n$ trials. This uses the combinatorics notation, where: $$ \binom{n}{x} = \frac{n!}{x!(n-x)!} $$
Don't panic! Your calculator handles this function (often labelled nCr).
$p^x$: This is the probability of getting $x$ successes.
$(1-p)^{n-x}$: This is the probability of getting the remaining $(n-x)$ failures.

3.2 Step-by-Step Example (Using the Formula)

Suppose you throw a biased coin 5 times, and the probability of getting a Head (Success) is $p=0.4$. We want to find the probability of getting exactly 3 Heads ($X=3$).

Here, $n=5$, $x=3$, $p=0.4$, and $1-p = 0.6$.

Step 1: Calculate the number of ways ($\binom{n}{x}$).
$$ \binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{120}{6 \times 2} = 10 $$ (There are 10 unique sequences that give 3 Heads and 2 Tails, e.g., HHHTT, HHTHT, etc.)

Step 2: Calculate the probability of the successes and failures.
$$ p^x (1-p)^{n-x} = (0.4)^3 (0.6)^{5-3} = (0.064)(0.36) = 0.02304 $$

Step 3: Multiply the results.
$$ P(X=3) = 10 \times 0.02304 = 0.2304 $$

3.3 Using Cumulative Probability Tables

Often, exam questions ask for cumulative probabilities (e.g., the probability of getting at most 5 successes). Cumulative tables provide $P(X \le x)$ for various values of $n$ and $p$.

Key Translations for Using Tables:

$P(X \le x)$: Read directly from the table.
$P(X < x)$: This is the same as $P(X \le x-1)$. (If less than 5, it means 4 or fewer.)
$P(X > x)$: This is calculated as $1 - P(X \le x)$. (If more than 5, it means 1 minus 5 or fewer.)
$P(X \ge x)$: This is calculated as $1 - P(X \le x-1)$. (If 5 or more, it means 1 minus 4 or fewer.)
$P(a \le X \le b)$: This is calculated as $P(X \le b) - P(X \le a-1)$.

Common Mistake to Avoid: When calculating $P(X > x)$ or $P(X \ge x)$, be very careful with the endpoint! For discrete variables, $P(X > 5)$ is NOT the same as $1 - P(X \le 5)$; it is $1 - P(X \le 4)$.

4. Mean, Variance, and Standard Deviation of the Binomial Distribution

Calculating the mean and variance for a Binomial distribution is much simpler than calculating these values for general discrete variables (where you sum $x \cdot P(x)$ or $(x-\mu)^2 \cdot P(x)$).

Since the Binomial distribution is just the sum of $n$ independent Bernoulli trials, we can simply add up the means and variances of those $n$ trials.

Recall the Bernoulli trial mean is $p$ and variance is $p(1-p)$.

4.1 Mean (Expected Value)

The expected number of successes, $E(X)$, is simply the number of trials multiplied by the probability of success in each trial.

$$E(X) = np$$

Example: If you take 20 shots at goal, and you score 30% of the time ($p=0.3$), you expect to score $E(X) = 20 \times 0.3 = 6$ goals.

4.2 Variance and Standard Deviation

The variance of the total number of successes is the number of trials multiplied by the variance of a single Bernoulli trial.

Variance: $$Var(X) = np(1 - p)$$

This is often written as: $$Var(X) = npq$$

Standard Deviation ($\sigma$): $$\sigma = \sqrt{np(1 - p)}$$

Key Takeaway: The formulas for the mean and variance of a Binomial distribution are provided by the syllabus as deductions from the Bernoulli distribution, so you must be able to use them confidently! They save a huge amount of time compared to standard discrete variable calculations.

Quick Review: Bernoulli vs. Binomial

Feature	Bernoulli Distribution	Binomial Distribution
Modelled Event	Single trial	Fixed number ($n$) of independent trials
Notation	Not usually given specific notation	$X \sim B(n, p)$
Mean	$E(X) = p$	$E(X) = np$
Variance	$Var(X) = p(1-p)$	$Var(X) = np(1-p)$