AQA AS Level Mathematics Paper 2, June 2025: Question 16

Mark scheme

Show the mark scheme

$Mark scheme for Question 16: M1 is awarded for summing the five probabilities and equating the sum to 1, giving 10p + 0.25 = 1. A1 is awarded for solving to find p = 0.075 (or an equivalent fraction). Total marks: 2.$

Q Marking instructions AO Marks Typical solution

16 Sums the probabilities and 1.1a M1

equates to 1. Allow one slip in 0.2 – p + 0.05 + 3p + 4p + 2p + 2p

the summing of the probabilities = 1

or the omission of one of the

probabilities. 10p + 0.25 = 1

PI by correct answer

10p = 0.75

Obtains 0.075 1.1b A1 p = 0.075

ACF

Question 16 Total 2

How to answer it

Finding an Unknown Constant in a Probability Distribution

What this question tests

Fundamental Law of Probability Distributions: The total sum of all probabilities across the entire sample space equals exactly 1 (i.e. Σ P(X = x) = 1).
Linear Algebraic Manipulation: Grouping like terms (collecting constants and algebraic terms in p) and solving a basic linear equation.
Accuracy & Forms: Accurately expressing final answers in decimal or exact fractional form.

Question 16 • [2 Marks]

Discrete Random Variable X

AQA AS Mathematics – Statistics

The table shows the probability distribution for a discrete random variable X:

x	0	1	2	3	4
P(X = x)	0.2 − p	0.05 + 3p	4p	2p	2p

Find the value of the constant p.

📐 Step-by-Step Calculation

Apply the total probability rule:
Σ P(X = x) = 1
(0.2 − p) + (0.05 + 3p) + 4p + 2p + 2p = 1
Collect constant terms:
0.2 + 0.05 = 0.25
Collect terms in p:
−p + 3p + 4p + 2p + 2p = 10p
Form and solve the simplified linear equation:
10p + 0.25 = 1
10p = 1 − 0.25
10p = 0.75
Divide to isolate p:
p = 0.75 ÷ 10 = 0.075 (or in fraction form: 3/40)

✅ Correct Answer

p = 0.075 or ³/₄₀

Mark Scheme Breakdown:

M1 (AO 1.1a): Sums probabilities and equates to 1. (Allows 1 arithmetic slip or 1 omitted term; implied by correct final answer).
A1 (AO 1.1b): Correct value of 0.075 (Any Correct Form: ACF).

💡 Key Knowledge

For any valid discrete probability distribution of a random variable X:
Σ P(X = x) = 1
Each individual probability must also satisfy:
0 ≤ P(X = x) ≤ 1
You can always check your answer by substituting p = 0.075 back into each column to confirm all values are positive and sum to 1.

🧠 Exam Technique & Examiner Insight

Write the full sum clearly: Always write down the unsimplified equation Σ P(X = x) = 1 explicitly to guarantee the M1 mark, even if you make an arithmetic error later.
Sanity check your values: Substitute p back into the first probability: 0.2 − 0.075 = 0.125 . Since 0.125 > 0 and ≤ 1, this confirms the probability is valid. If your p gave a negative probability, you'd immediately know a mistake was made.

❌ Common Errors to Avoid

Sign error with the first term: Overlooking the minus sign in −p , leading to 12p instead of 10p .
Omission of a column: Forgetting one of the identical 2p terms at x = 3 or x = 4 when writing out the sum.
Arithmetic misstep dividing by 10: Moving the decimal point the wrong way (e.g. giving 0.75 or 7.5 instead of 0.075).
Confusing x with P(X = x): Adding values of x (0 + 1 + 2 + 3 + 4) instead of their corresponding probabilities.

Topics

Statistics · N: Statistical distributions