IB Maths AA probability: conditional probability, binomial and normal

What probability covers at SL and HL

This follows the current Analysis and Approaches guide (first assessment 2021). The same topic also includes statistics such as regression and summary statistics, which are not covered on this page.

The key ideas

Conditional probability is the heart of the topic. P(A|B) is the probability of A given that B has happened, so you restrict the sample space to B: P(A|B) = P(A n B)/P(B). Two events are independent if P(A|B) = P(A), which is equivalent to P(A n B) = P(A) × P(B). Mutually exclusive is different: it means P(A n B) = 0, and two events with non-zero probabilities cannot be both.

A random variable assigns a number to each outcome. For a discrete variable the probabilities must add to 1, and the expected value is E(X) = sum of x × P(X = x). A game is fair when the expected gain is zero.

The binomial distribution counts successes in n independent trials, each with the same probability p. The normal distribution models continuous data that cluster symmetrically around a mean. For both, the GDC does the arithmetic, but you must state the distribution and parameters, and choose the right command: a probability of an exact value, a cumulative probability, or an inverse normal.

Worked example 1: conditional probability and a tree without replacement

1. Given P(A) = 0.5, P(B) = 0.4 and P(A n B) = 0.2: P(A|B) = 0.2/0.4 = 0.5, which equals P(A), so A and B are independent. Check: P(A) × P(B) = 0.2. Also P(A u B) = 0.5 + 0.4 - 0.2 = 0.7.
2. A bag holds 5 red and 3 blue counters. Two are taken without replacement. P(both red) = 5/8 × 4/7 = 20/56 and P(both blue) = 3/8 × 2/7 = 6/56.
3. P(same colour) = 20/56 + 6/56 = 26/56 = 13/28.
4. P(first is red | same colour) = P(both red)/P(same colour) = (20/56)/(26/56) = 10/13.
5. Check: P(different colours) = 5/8 × 3/7 + 3/8 × 5/7 = 30/56, and 26/56 + 30/56 = 1.

Worked example 2: a binomial distribution

1. A student guesses 10 multiple-choice questions, each with probability 0.3 of being right. Let X be the number correct, so X ~ B(10, 0.3).
2. P(X = 2) = C(10, 2) × 0.3^2 × 0.7^8 = 45 × 0.09 × 0.0576... = 0.233 (3 s.f.). On the GDC this is binompdf(10, 0.3, 2).
3. P(X <= 2) = binomcdf(10, 0.3, 2) = 0.383 (3 s.f.). Check by adding: P(X = 0) = 0.0282, P(X = 1) = 0.1211, P(X = 2) = 0.2335, total 0.3828.
4. P(X >= 3) = 1 - P(X <= 2) = 0.617. 'At least' questions are always easier through the complement.
5. E(X) = np = 3 and Var(X) = np(1 - p) = 10 × 0.3 × 0.7 = 2.1.

Worked example 3: the normal distribution and expected value

1. Heights H of a group are modelled by H ~ N(170, 8^2) in cm. P(H > 180): the z-value is (180 - 170)/8 = 1.25, and the GDC gives P(H > 180) = 0.106 (3 s.f.).
2. Find h so that 90% of the group are shorter than h. Inverse normal with area 0.9 gives z = 1.2816, so h = 170 + 1.2816 × 8 = 180 cm (3 s.f.).
3. Check: 180 cm should be close to the 90th percentile, and the first part showed that about 10.6% are taller than 180, which agrees.
4. Expected value: a discrete variable takes the values 0, 1 and 2 with probabilities k, 2k and 3k. The probabilities add to 1, so 6k = 1 and k = 1/6.
5. E(X) = 0 × 1/6 + 1 × 2/6 + 2 × 3/6 = 8/6 = 4/3. Check: the answer lies between 0 and 2 and nearer 2, where most of the probability is.

Common mistakes that cost marks

• Not defining the random variable or its distribution. Writing X ~ B(10, 0.3) often earns a mark on its own.
• Treating 'without replacement' as independent trials.
• Confusing independent with mutually exclusive.
• Using P(X < 3) when the question says 'at most 3'. For a discrete variable these are different.
• Entering the variance instead of the standard deviation into the normal command.
• Rounding intermediate probabilities to two decimal places, which pushes the final answer off.
• Writing calculator syntax, such as normalcdf(...), as your only working. Write the probability statement in proper notation too.

Exam technique and how a tutor helps

Start every probability question by writing what the event is in words or symbols. Draw a Venn or tree diagram whenever there is more than one event, because many marks are for correctly filled diagrams. On Paper 1 the numbers are chosen to work by hand, so expect fractions; on Paper 2 expect the GDC and answers to three significant figures unless the question says otherwise.

Watch the wording: 'given that' signals conditional probability, 'at least' suggests the complement, and 'expected' means E(X). In multi-part questions a later part often uses a distribution set up earlier, so keep exact or full calculator values.

In one-to-one lessons a tutor asks the student to say which distribution applies and why before any calculation, which fixes the most common error at the root. Lessons alternate short GDC drills, so the binomial and normal commands become automatic, with mixed past-paper questions where the student must decide the method. HL students also practise continuous random variables, where probability meets integration.

Self-check: can you do these?

• P(A) = 0.6, P(B) = 0.5 and P(A u B) = 0.8. Find P(A n B) and P(B|A). (Answer: 0.3 and 0.5)
• X ~ B(8, 0.5). Find P(X = 4). (Answer: 70/256 = 0.273 to 3 s.f.)
• X ~ B(20, 0.25). Find E(X) and Var(X). (Answer: 5 and 3.75)
• Z ~ N(0, 1). Find P(-1 < Z < 1). (Answer: 0.683 to 3 s.f.)
• A fair die is rolled. What is P(score > 4 | score is even)? (Answer: 1/3)