IB Maths AA: Probability and Expectation - Tutor Lesson Plan (independence, conditional probability, variance)

IB Maths AA: Probability and Expectation – Tutor Lesson Plan (independence, conditional probability, variance)

2026-11-05

Students know that independent events multiply, so when a question says given B occurred they multiply anyway. One value of 0.21 carries three different questions here.

Objectives — the student can decide when multiplication is allowed, compute a conditional probability as the intersection over the given event, and get the mean and variance from a probability table.

01 Prerequisites (3-minute check)

If any of these fails, fix it before the lesson starts.

Skill	Quick check	Pass mark
Multiplying decimals	What is 0.4 times 0.7?	0.28
Fraction to decimal	What is 21 divided by 70?	0.3
Averages	The mean of 0, 1, 2 and 3	1.5

02 Lesson flow (60 minutes)

Time	Stage	Tutor does	Student does
0-8 min	Three question types	Both, given, and expected value	State how they differ
8-20 min	Independent events	P(A) 0.4, P(B) 0.7	Get 0.28
20-34 min	Conditional probability	P(A and B) 0.21, P(B) 0.7	Get 0.3
34-48 min	Expectation	probabilities 0.2, 0.3, 0.4, 0.1 over 0 to 3	Get 1.4
48-56 min	Variance	E(X squared) minus E(X) squared	Get 0.84
56-60 min	Exit check	Three questions marked	Finish alone

03 Teaching script (read it out)

Scene 1: Multiply only after independence is established

For independent A and B, P(A and B) = 0.4 times 0.7 = 0.28.
Worked the other way: a question that gives P(A and B) = 0.21 while P(A)P(B) = 0.28 has already shown the events are not independent.

Student judges what the gap between 0.21 and 0.28 means (the events are linked).

Scene 2: Conditional probability changes the denominator

P(A given B) = P(A and B) divided by P(B) = 0.21 divided by 0.7 = 0.3.
The usual slip is dividing by 0.4. The denominator is always the event that is known to have happened, not the one being asked about.

Student explains why the denominator is 0.7.

Scene 3: Expectation and variance from one table

E(X) = 0(0.2) + 1(0.3) + 2(0.4) + 3(0.1) = 1.4.
E(X squared) = 0 + 0.3 + 1.6 + 0.9 = 2.8, so Var = 2.8 – 1.96 = 0.84.

Note: variance is not E(X squared); the square of the mean must be subtracted.

04 Question types and worked steps

Type	Example	Steps	Answer
Independent events	P(A) 0.4, P(B) 0.7	Multiply directly	0.28
Conditional probability	P(A and B) 0.21, P(B) 0.7	Intersection over the given event	0.3
Expectation	table over 0 to 3	Sum of x times P(x)	1.4
Variance	E(X squared) 2.8, E(X) 1.4	E(X squared) minus E(X) squared	0.84

Sanity check (teach this): The probabilities must sum to 1 (here 0.2 + 0.3 + 0.4 + 0.1). If they do not, a row is missing and the expectation is already wrong.

05 Independent practice (3 questions)

Question	Answer	What to watch
Probability of both, independent	0.28	Independence checked first
Value of P(A given B)	0.3	Denominator is 0.7, not 0.4
Mean and variance	mean 1.4, variance 0.84	The mean squared was subtracted

06 Common mistakes and what to say

Mistake	Student writes	Say this
Wrong denominator	answer of 0.525	The denominator is the event known to have happened.
Multiplying dependent events	answer of 0.28	The question already gives 0.21, so 0.28 contradicts it.
Variance left as E(X squared)	variance of 2.8	Variance equals E(X squared) minus the mean squared.
Probabilities not summing to 1	table totals 0.9	Check the total before computing anything.

07 Homework (paper, section and question numbers)

IB maths chapter index — Use the three questions on this page

08 Exit check (last 5 minutes)

Question	Answer	Decision
P(A and B) when independent	0.28	All three right: move to the normal distribution
P(A given B)	0.3	One wrong: re-teach conditional probability
Mean and variance	1.4 and 0.84	Two wrong: repeat scene 3

09 Tutor FAQ

Are probability formulas in the booklet?

Some are, but the definition of conditional probability must be reproduced.

Variance or standard deviation?

The standard deviation is the square root, in the same units as the data.

Memorise the table method?

No, but always check that the probabilities sum to 1.

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A TutorZone lesson-prep guide. Written by the TutorZone team; questions and lesson flow are original and are not taken from any school or publisher material.