Probability and distributions practice questions (A-Level Maths)

Sample questions

The one free A-Level Maths question on this topic, from the free A-Level Maths Practice Paper. Each answer is behind its own toggle; the whole paper, its PDF and the full answer key are on the exam page.

1. Question 18 marks · Discrete random variables & the binomial distribution
   Statistics. The discrete random variable $X$ has the probability distribution $P(X = 1) = 0.1$, $P(X = 2) = 0.3$, $P(X = 3) = k$, $P(X = 4) = 0.2$, where $k$ is a constant.
   $P(X=1) = 0.1, \quad P(X=2) = 0.3, \quad P(X=3) = k, \quad P(X=4) = 0.2$
   (a)

   Find the value of $k$.

   [1]
   (b)

   Find $E(X)$ and $\text{Var}(X)$.

   [4]
   (c)

   Five independent observations of $X$ are taken. Find the probability that exactly two of them take the value 4.

   [3]
   Show answerFrom the free A-Level Maths Practice Paper →

Key terms

The definitions and formulas A-Level Maths expects for probability and distributions, typeset as they appear on the paper.

Addition rule and mutually exclusive events

The probability of $A$ or $B$ subtracts the overlap counted twice. Mutually exclusive events have no overlap, so the probabilities simply add.

$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$

Conditional probability and independence

The probability of $A$ given $B$ restricts to the outcomes in $B$. Events are independent when the conditional probability equals the unconditional one; check with the product rule.

$$P(A \mid B) = \frac{P(A \cap B)}{P(B)}, \qquad \text{independent} \iff P(A \cap B) = P(A)P(B)$$

Discrete random variable

A variable taking separate values each with a probability; the probabilities sum to 1, which is how an unknown constant in a distribution table is found. The discrete uniform distribution gives equal probabilities.

$$\sum P(X = x) = 1$$

Binomial distribution

The number of successes in $n$ independent trials with constant probability $p$. State the conditions when asked why the model is suitable, and define the variable with its parameters before calculating.

$$X \sim B(n, p), \qquad P(X = r) = \binom{n}{r}p^r(1 - p)^{n-r}, \qquad E(X) = np$$

Normal distribution

The continuous bell curve, symmetric about $\mu$, with about 68 percent of values within one standard deviation and 95 percent within two. Standardise to find probabilities or to solve for an unknown $\mu$ or $\sigma$ from a given probability.

$$X \sim N(\mu, \sigma^2), \qquad Z = \frac{X - \mu}{\sigma} \sim N(0, 1)$$

Normal approximation to the binomial

For large $n$ and $p$ close to 0.5, the binomial is approximately normal with the same mean and variance; apply a continuity correction of 0.5 when switching from discrete to continuous.

$$B(n, p) \approx N\big(np,\; np(1 - p)\big)$$

Correlation and regression

The product moment correlation coefficient $r$ measures linear correlation between $-1$ and $1$. The regression line of $y$ on $x$ predicts $y$ from $x$ (the explanatory variable); interpolation is reliable, extrapolation is not.

$$y = a + bx$$

Frequently asked questions

Other A-Level Maths topics

• Proof, algebra and functions
• Coordinate geometry
• Sequences and series
• Trigonometry
• Exponentials and logarithms
• Differentiation
• Integration
• Numerical methods and vectors
• Sampling and data presentation
• Hypothesis testing
• Kinematics
• Forces, Newton's laws and moments

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