The Binomial Distribution (AQA A-Level Mathematics): Revision Notes

Example: Consider a game in which a dice is rolled $20$ $times$. Let $X$ be the number of "sixes" rolled in total.

• Consider $X$ to be the number of "sixes".
• We have $P(\text{win}) = \frac{1}{6}$.

Thus,

In the previous scenario, find the probability we roll exactly seven 6s in the twenty throws.

Explanation:

• Seven $6s$
• Seven wins
• Thirteen losses
• Number of different orders events can happen in

Example: The random variable $X \sim B(8, \frac{1}{3})$.

Questions:

Find:

a) $P(X = 2)$

b) $P(X = 5)$

c) $P(X \leq 1)$

1. Find $P(X = 2)$ Step 1: Use the Binomial Probability Formula

For a binomial distribution, the probability of getting r successes in n trials is given by:

Where:

• $n = 8$ (the number of trials)
• $p = \frac{1}{3}$ (the probability of success on each trial)
• $r = 2$ (the number of successes we want)

Step 2: Substitute the Values and Calculate the Result

Instructions for Calculator:

2. Go to "$Dist$" by pressing [$SHIFT$] then [$1$].
3. Select [$4:Binomial$ $PD$].
4. Enter $X = 2, n = 8, p = 1/3$.
5. Calculate $P = :success[0.2731290962]$.

b. Find $P(X=5)$

c. Find $P(X \leq 1)$

Step 1: Understand the Question

We are asked to find the probability that X is less than or equal to 1.

This can be shown in this notation:

Step 2: Use the Binomial Formula for $P(X=0)$ and $P(X=1)$

For $P(X=0)$

For $P(X=1)$

Step 3: Calculate the Result

Example: The probability of a switch being faulty is $0.08$. A random sample of $10$ switches is taken from the production line.

Questions:

a) Define a suitable distribution to model the number of faulty switches in this sample, and justify your choice.

b) Find the probability that the sample contains $4$ faulty switches.

a) Define a suitable distribution to model the number of faulty switches in this sample, and justify your choice.

Step 1: Identify the Distribution

We are asked to model the number of faulty switches in a random sample of $10$ switches, with the probability of any switch being faulty given as $0.08$.

This is a binomial distribution because:

• There is a fixed number of trials ($10$ switches)
• Each switch is either faulty or not faulty (a binary outcome)
• The probability of a switch being faulty is constant at $0.08$
• The switches are independent of each other. Thus, the number of faulty switches X follows the binomial distribution:

Where:

• $10$ is the number of trials (switches)
• $0.08$ is the probability of success (faulty switch)

Step 2: Justify the Choice

• The trials (checking each switch) are independent because whether one switch is faulty does not affect the others.
• The probability of finding a faulty switch remains constant at $0.08$ for each switch.

b) Find the probability that the sample contains 4 faulty switches.

Step 1: Use the Binomial Formula

We need to calculate $P(X=4)$ where $X \sim B(10, 0.08)$

Where:

• $n = 10$ (the number of switches)
• $p = 0.08$ (the probability of a switch being faulty)
• $r = 4$ (the number of faulty switches)

Step 2: Substitute the Values and Calculate the Result

Thus, the probability that the sample contains $4$ faulty switches is approximately $0.005216$ or $5.216 \times 10^{-3}$