Very frequent

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Probability & Statistics

Conditional probability, Bayes, distributions, and the summary statistics that Round 1 asks you to compute by hand.

Prerequisites

Foundations - Mathematical Logic & Reasoning

Round 1's fill-in-the-blank questions are worth five marks each and demand an exact answer. Probability is a common source of them, because the answers are unambiguous numbers.

Conditional probability and Bayes

P(A \mid B) = \frac{P(A \cap B)}{P(B)}, \qquad P(A \mid B) = \frac{P(B \mid A)P(A)}{P(B)}

Expand the denominator by total probability:

P(B) = P(B \mid A)P(A) + P(B \mid \neg A)P(\neg A)

The classic question: a test is 99% sensitive and 95% specific for a condition with 1% prevalence; a patient tests positive. Then

P(\text{disease} \mid +) = \frac{0.99 \times 0.01}{0.99 \times 0.01 + 0.05 \times 0.99} \approx 0.167

Under 17%, against an intuition that says 99%. The base rate is doing all the work, and recognising that instantly is what the question tests.

Independence

$A$ and $B$ are independent when $P(A \cap B) = P(A)P(B)$ . Independence and mutual exclusivity are different and often confused: mutually exclusive events with non-zero probability are dependent, because knowing one occurred tells you the other did not.

Distributions worth knowing

| Distribution | Mean | Variance | Use | | --- | --- | --- | --- | | Bernoulli( $p$ ) | $p$ | $p(1-p)$ | One trial | | Binomial( $n,p$ ) | $np$ | $np(1-p)$ | Count of successes | | Poisson() | | | Rare events per interval | | Normal() | | | Continuous, sums of many effects |

The Poisson having equal mean and variance is a common single-mark question.

Summary statistics by hand

Mean, median, mode, variance and standard deviation on a small dataset, computed quickly and correctly. Remember the two variance formulas differ:

\sigma^2 = \frac{1}{n}\sum(x_i - \bar x)^2 \quad \text{(population)}, \qquad s^2 = \frac{1}{n-1}\sum(x_i - \bar x)^2 \quad \text{(sample)}

Using $n$ where the question wanted $n-1$ produces a wrong exact answer, which on a fill-in-the-blank is worth zero. Read which one is asked for.

Expectation and variance rules

\mathbb{E}[aX + b] = a\mathbb{E}[X] + b, \qquad \operatorname{Var}(aX+b) = a^2\operatorname{Var}(X)

The $a^2$ is the detail people drop, and the constant $b$ vanishing from the variance is the other half of the same question.

Expand the Bayes denominator by total probability; the base rate drives the answer. Mutually exclusive is not independent - with non-zero probabilities it is the opposite.

Check whether a variance question wants the population or sample formula. Variance scales by $a^2$ and ignores additive constants.

Problem

Shifting a variance

introwritten

$\mathrm{Var}(X) = 5$ . What is $\mathrm{Var}(X + 12)$ ?

Show solution

An additive constant does not enter the variance. The answer is still 5.

Your answer

Problem

Two heads

introwritten

Two fair coins are tossed independently. What is $P(\text{both heads})$ ?

Show solution

Independence is the product: $(1/2)\times(1/2) = 1/4$ .

Select an answer

A1/2
B1/3
C1/4
D1

Probability & Statistics

Prerequisites

Probability & Statistics

Prerequisites

Checkpoint

12 questions, 48 marks, 30:48 at the paper's pace.

Resources

Past paper problems

Conditional probability and Bayes

Independence

Distributions worth knowing

Summary statistics by hand

Expectation and variance rules

Shifting a variance

Two heads