Should I memorise formulae or draw a table for IB probability?

Draw the sample space first. Two dice give only 36 ordered outcomes, and once the table is written the denominator appears by itself; formulae merely compress the table and invite double counting.

Does the expected value mean the outcome most likely to happen?

No. The expected value is a long-run average and need not be a possible outcome at all: a single die has expectation 3.5, which can never be rolled.

IB Mathematics: Probability and Expectation (Challenge, 10 Questions)

2026-11-06

Tutor Prep Guide · IB Mathematics

Ten challenge items on sampling without replacement, geometric probability, expected net gain, the standard deviation of a binomial, E(X squared) and the birthday problem, each needing full reasoning with complements or the variance identity.

Quick answer Answers in order: 5/14, 5/8, 23/28, 0.2826, -$4, E = 6 with SD about 2.05, 0.6, 329/6 or about 54.83, 17/72 or about 0.236, option B by $4.

01 Paper details (check answers afterwards)

Item	Detail
Questions	10 questions
Suggested time	about 45 minutes
Difficulty mix	Challenge 10
Chapter	Chapter: IB Mathematics: Probability and Expectation
Level	IB

02 Questions (10 items)

ChallengeA bag holds 5 red and 3 white balls. Two are drawn without replacement. Find the probability that both are red.
ChallengeFrom the same bag of 5 red and 3 white balls, two are drawn without replacement. Find the probability that the second ball is red.
ChallengeFrom the same bag, three balls are drawn without replacement. Find the probability that at least one is white.
ChallengeA point is chosen at random inside a square of side 10 cm. Find the probability that it lies inside a circle of radius 3 cm (take pi = 3.14).
ChallengeA raffle costs $8 per ticket, wins $200 with probability 1/50. Find the expected net gain per ticket.
ChallengeX is binomial with n = 20 and p = 0.3. Find E(X) and the standard deviation to 2 decimal places.
ChallengeA and B are independent with P(A) = 0.5 and P(A union B) = 0.8. Find P(B).
ChallengeX is the total of two fair dice. Find E(X squared).
ChallengeThree people are chosen at random. Find the probability that at least two share a birth month, assuming 12 equally likely months.
ChallengeOption A pays $20 for certain; option B pays $80 with probability 0.3 and nothing otherwise. Which has the higher expectation, and by how much?

03 Answers and one-line notes

#	Answer	One-line note
1	5/14	(5/8) x (4/7) = 20/56 = 5/14; the denominator drops by one after the first draw.
2	5/8	By symmetry the second draw has the same marginal probability as the first, namely 5/8, with no case split needed.
3	23/28	All red is (5/8)(4/7)(3/6) = 5/28, so the answer is 1 – 5/28 = 23/28.
4	0.2826	The circle has area 9 x 3.14 = 28.26, which over the square area of 100 gives 0.2826.
5	-$4	The expected prize is 200/50 = $4, and 4 – 8 = -4 dollars.
6	E(X) = 6, SD is about 2.05	E = np = 6 and Var = np(1 – p) = 4.2, so the standard deviation is the square root of 4.2, about 2.05.
7	0.6	Independence gives 0.5 + q – 0.5q = 0.8, so 0.5q = 0.3 and q = 0.6.
8	329/6, about 54.83	The variance is 2 x 35/12 = 35/6, so E(X squared) = 35/6 + 49 = 329/6.
9	17/72, about 0.236	All different is (12 x 11 x 10)/1728 = 1320/1728, so the answer is 1 – 1320/1728 = 17/72.
10	Option B, by $4	Option B has expectation 0.3 x 80 = $24, which is $4 above option A.

04 What to do after marking

After marking, redo only the wrong items and write down which step went wrong (formula, unit, or misread number).
Repeat the same item two days later; move to the next paper only when both attempts are correct.
Parents only need the answer and note columns; there is no need to explain every item.

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