STEP probability and statistics: guide and practice problems

STEP probability questions are about careful counting and clear arguments, often with a neat idea such as symmetry or linearity of expectation.

Where it comes from: Statistics in A Level Mathematics; continuous random variables in Further Statistics.

Key ideas

Games that repeat: the probability is a geometric series, or set up an equation using 'what happens after one round'.
Linearity of expectation works even when events are not independent: count with indicator variables.
Symmetry: every ball is equally likely to be in any position.
For a continuous random variable, use the pdf to find k first, then the mean and variance; the maximum of independent copies uses products of the cdf.

Common traps

Assuming independence that the question did not give.
Using the variance formula with E(X)^2 and E(X^2) swapped.
Answers above 1 for a probability: always sanity-check.

Practice problems

Original problems in the style of STEP. Try each one for at least 20 minutes before you open a hint; write a full solution, then compare.

Problem 1: First to throw a head · STEP 2 style

Three players \(A\), \(B\), \(C\) take turns, in that order, to toss a coin that shows heads with probability \(p\) (\(0

Show that \(A\) wins with probability \(\dfrac{p}{1-(1-p)^3}\), and find the corresponding probabilities for \(B\) and \(C\).
For a fair coin, find all three probabilities.
Find the expected total number of tosses in a game, in terms of \(p\).

Hint 1

\(A\) wins on toss 1, 4, 7, \(\dots\): a geometric series with ratio \((1-p)^3\).

Hint 2

\(B\)'s chance is \((1-p)\) times \(A\)'s, and \(C\)'s is \((1-p)^2\) times \(A\)'s. The number of tosses is geometric.

Full solution

(i) With \(q=1-p\): \(P(A)=p+q^3p+q^6p+\cdots=\dfrac{p}{1-q^3}\). \(B\) can only win once \(A\) has missed, after which \(B\) is "first", so \(P(B)=\dfrac{qp}{1-q^3}\), and \(P(C)=\dfrac{q^2p}{1-q^3}\).

(ii) \(p=q=\tfrac12\): \(1-q^3=\tfrac78\), so \(P(A)=\tfrac47\), \(P(B)=\tfrac27\), \(P(C)=\tfrac17\).

(iii) The number of tosses until the first head is geometric with mean \(\dfrac1p\).

Results: Fair coin: \(\frac47,\frac27,\frac17\); expected tosses \(\frac1p\).

2 more probability and statistics problems

A parabolic density; Waiting for the first red. Each with two hints and a full solution.

Included with A Level plans, and with IB, IGCSE or CBSE plans.

Next steps

Related topics: Mechanics · Algebra and inequalities. Stuck? Read the problem-solving strategies. Ready for the real thing? Official STEP past papers.

STEP is run by OCR. A Level Math Revision is independent: it is not affiliated with or endorsed by OCR, the University of Cambridge or any university. Every problem here is our own, written in the style of STEP; for real papers use OCR's official past papers.