10 original competition-style problems: dice, cards, areas and expected values. Try each one before opening the hints; the second hint gives more away, and the full solution explains why the method works and where the idea leads.
Every question is free to read. The hints, answer checking and full solutions are included with every A Level, IB, IGCSE and CBSE plan. See plans.
Answer a problem to track what you have solved.
Press Project on any problem to show it full screen with a timer, the hints, the answer and the worked solution one step at a time (arrow keys move between problems; Space reveals the next step; F full screen; Esc closes). Switch on the ‘Add to worksheet’ buttons, pick problems, then print them from the worksheet builder or set them as homework for a class, with the full solutions as the mark scheme. Free problems are free for every class; problems marked ‘With a plan’ can be set by teachers with a plan or school licence. Ready-made sessions: maths club packs.
A fair coin is tossed 10 times. A run is a maximal block of consecutive equal results (HHTHHH has 3 runs). What is the expected number of runs?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Spot the pattern and generalise
The numbers 1 to 10 are arranged in a random order. A number is a record if it is larger than every number before it (the first number is always a record). What is the expected number of records? (Give an exact fraction.)
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Symmetry
Five cards are dealt from a well-shuffled standard pack of 52 (four suits of 13). What is the probability that all four suits appear among the five cards?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Pigeonhole principle, Organised cases
A random arrangement of 1, 2, …, 10 is chosen. Let X be the number of fixed points (numbers in their own position). What is the expected value of X2?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Symmetry
An ant starts at a corner of a cube and walks along the edges. At each corner it picks one of the three edges at random. What is the expected number of edges it walks before first reaching the opposite corner?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Symmetry, Working backwards
An urn starts with 1 red and 1 blue ball. Ten times, a ball is drawn at random and put back together with another ball of the same colour. What is the probability that exactly 5 of the 10 added balls are red?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Symmetry, Spot the pattern and generalise
Ten balls are dropped independently into ten boxes, each ball equally likely to land in any box. What is the expected number of empty boxes? (Give an exact fraction.)
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Count the opposite
Three job candidates, of different quality, are interviewed one at a time in a random order. After each interview you must hire or reject at once. Your strategy: reject the first candidate, then hire the first later candidate who is better than the first; if none is, hire the last. What is the probability that you hire the best candidate?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Organised cases
A fair die is rolled repeatedly until two consecutive rolls show the same number. What is the expected total number of rolls?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Working backwards
Two numbers are chosen independently and uniformly at random from [0, 1]. What is the expected distance between them?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Symmetry
More Olympiad-style problems: Number theory · Combinatorics · Geometry · Algebra · Logic · Calculus and functions
Probability at other levels: Junior (ages 11 to 13) · Intermediate (ages 13 to 16) · Senior (ages 16 to 18)
Problem-solving strategies · Where next · Extension & competition maths
A Level Math Revision