What is the difference between independent and dependent events in Further Probability?

In Further Probability, independent events are those whose outcomes do not affect each other. For example, flipping a coin and rolling a die are independent events. Dependent events, on the other hand, are those where the outcome of one event affects the outcome of another. An example would be drawing two cards from a deck without replacement, where the outcome of the first draw affects the probability of the second.

How do you calculate conditional probability in the IB DP Mathematics curriculum?

Conditional probability is calculated using the formula P(A|B) = P(A and B) / P(B), where P(A|B) is the probability of event A occurring given that event B has occurred. This concept is crucial in the IB DP Mathematics curriculum as it helps students understand how probabilities change when additional information is known.

What is Bayes' Theorem and how is it applied in Further Probability?

Bayes' Theorem is a fundamental concept in Further Probability that describes how to update the probability of a hypothesis based on new evidence. It is expressed as P(A|B) = [P(B|A) * P(A)] / P(B). This theorem is particularly useful in the IB DP Mathematics curriculum for solving problems involving reverse conditional probabilities, such as medical testing or decision-making scenarios.

Can you explain the concept of mutually exclusive events in Further Probability?

Mutually exclusive events are events that cannot occur simultaneously. In Further Probability, if two events A and B are mutually exclusive, the probability of both occurring at the same time is zero, i.e., P(A and B) = 0. This concept is important in the IB DP Mathematics curriculum as it helps students understand how to calculate probabilities when events do not overlap.

How is the concept of permutations and combinations used in Further Probability?

Permutations and combinations are used in Further Probability to calculate the number of ways events can occur. Permutations consider the order of events, while combinations do not. In the IB DP Mathematics curriculum, understanding these concepts is essential for solving problems involving arrangements and selections, such as determining the probability of specific outcomes in complex scenarios.

Why is "$\text{Var}(X - Y) = \text{Var}(X) - \text{Var}(Y)$." a common mistake in Further Probability?

Why it happens: Misapplying linearity to variance. How to avoid it: Variance is always non-negative — subtracting could go negative. The formula is PLUS regardless of sign.

Why is "$\text{Var}(3X) = 3\text{Var}(X)$." a common mistake in Further Probability?

Why it happens: Treating variance like expectation. How to avoid it: Variance scales by the SQUARE of the coefficient: Var(3X) = 9Var(X).

Why is "Using $\text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y)$ without checking independence." a common mistake in Further Probability?

Why it happens: Skipping the assumption check. How to avoid it: Always state 'X and Y are independent' before adding variances. Without it, Cov(X, Y) enters the formula.

ProbabilityIB Maths AA HL

Further Probability

Work through the notes, try the practice questions, then take the quiz. The report tells you exactly what to revise next.

PreviousNext

Learn this Topic

Start with the slides for the quick version, then go deeper with the full study notes.

Short Study Notes in the form of Slides

Read the notes first. If the method in a worked example clicks, you're ready for the questions.

Short Study Notes — Further Probability

Start with these resources to cover the key concepts, then work through the practice questions.

Page 1 / 0

Detailed Notes

Full prose, callouts and a recap — built for A* mastery, not just a quick scan.

Take these study notes with you

Download a branded PDF — full prose, callouts, recap and memorise list for Further Probability, ready to print or save offline.

Further Probability — IB Maths AA HL: linear combinations, expectation algebra, $E(aX + bY)$ and $\text{Var}(aX + bY)$

AA HL extends probability with algebra of expectation and variance for linear combinations of independent random variables — the bridge to inference. This note covers the rules and their use with binomial and normal sums.

What you’ll learn

Mapped to the IB DP Maths AA HL subject guide (2021 onwards (applies to 2026 exams)).

AO1 — State expectation and variance rules for linear combinations.
AO2 — Compute $E$ and $\text{Var}$ for $aX + bY$ .
AO2 — Apply normal sum/difference formulas.
AO3 — Recognise when independence is required.

Algebra of expectation and variance

Linearity always; variance only with independence.

Linearity of expectation (always, independent or not):

$E(aX + bY + c) = aE(X) + bE(Y) + c.$

Variance of a linear transformation (single variable, always):

$\text{Var}(aX + b) = a^2 \text{Var}(X).$

The $+b$ shifts but does not stretch.

Variance of a sum — INDEPENDENCE REQUIRED:

$\text{Var}(aX + bY) = a^2 \text{Var}(X) + b^2 \text{Var}(Y).$

Note the squared coefficients and that subtraction still adds variances: $\text{Var}(X - Y) = \text{Var}(X) + \text{Var}(Y)$ .

Worked example. $X$ has $E(X) = 5, \text{Var}(X) = 4$ . $Y$ has $E(Y) = 8, \text{Var}(Y) = 9$ . $X, Y$ independent. Find $E(2X - 3Y + 1)$ and $\text{Var}(2X - 3Y + 1)$ .

$E = 2(5) - 3(8) + 1 = 10 - 24 + 1 = -13$ .

$\text{Var} = 4(4) + 9(9) = 16 + 81 = 97$ . (constant $+1$ contributes 0)

Mean: linearity ALWAYS holds.
Variance: squared coefficients AND independence.
$\text{Var}(X - Y) = \text{Var}(X) + \text{Var}(Y)$ (PLUS!).

Sums and differences of normals

Normals stay normal.

If $X \sim N(\mu_X, \sigma_X^2)$ , $Y \sim N(\mu_Y, \sigma_Y^2)$ are independent:

$aX + bY \sim N(a\mu_X + b\mu_Y, a^2\sigma_X^2 + b^2\sigma_Y^2).$

Worked example. $X \sim N(60, 25)$ (mass of bag), $Y \sim N(58, 16)$ . Independent. Find $P(X > Y)$ .

Let $D = X - Y$ . Then $D \sim N(60 - 58, 25 + 16) = N(2, 41)$ , so $\sigma_D = \sqrt{41} \approx 6.403$ .

$P(X > Y) = P(D > 0) =$ normCdf $(0, 10^{99}, 2, 6.403) \approx 0.624$ .

Normal sums are normal — no need for limit theorems.
Variances ADD, never subtract.
State the resulting distribution before using GDC.

How it’s examined

Paper 1: short expectation/variance manipulations. Paper 2: normal sum/difference probabilities via GDC. Paper 3: extended applications with multiple random variables.

Sources: IB Diploma Programme Mathematics: Analysis and Approaches subject guide (IBO, official). Last reviewed 2026-05-31.

Master this Topic

Worked examples, formulae, definitions and the mistakes examiners flag — everything you need to push from a pass to an A*.

Take this whole topic with you

Download a branded revision sheet — worked examples, formulae, definitions and common mistakes for Further Probability, ready to print or save as PDF.

Step-by-step worked examples — Further Probability

Step-by-step solutions to past-paper-style questions on further probability, written exactly the way a tutor would explain them at the board.

1Linear transformation of one variable
Getting started• expectation
Question
$E(X) = 10, \text{Var}(X) = 4$ . Find $E(3X - 5)$ and $\text{Var}(3X - 5)$ .
Step-by-step solution
1. Step 1
  $E(3X - 5) = 3(10) - 5 = 25$ .
2. Step 2
  $\text{Var}(3X - 5) = 9 \text{Var}(X) = 9(4) = 36$ .
Answer
$E = 25$ , $\text{Var} = 36$ .
2Sum of two independent variables
Building confidence• independence
Question
$X, Y$ independent. $E(X) = 4, E(Y) = 7, \text{Var}(X) = 9, \text{Var}(Y) = 16$ . Find $E(2X + 3Y)$ and $\text{Var}(2X + 3Y)$ .
Step-by-step solution
1. Step 1
  $E(2X + 3Y) = 2(4) + 3(7) = 8 + 21 = 29$ .
2. Step 2
  $\text{Var}(2X + 3Y) = 4 \text{Var}(X) + 9 \text{Var}(Y) = 4(9) + 9(16) = 36 + 144 = 180$ .
Answer
$E = 29$ , $\text{Var} = 180$ .
3Difference of normals
Stretch• normal, GDC
Question
$X \sim N(100, 36), Y \sim N(95, 49)$ independent. Find $P(X > Y)$ .
Step-by-step solution
1. Step 1
  $D = X - Y$ . $E(D) = 100 - 95 = 5$ .
2. Step 2
  $\text{Var}(D) = 36 + 49 = 85$ , $\sigma_D = \sqrt{85} \approx 9.22$ .
3. Step 3
  $D \sim N(5, 85)$ . $P(X > Y) = P(D > 0)$ .
4. Step 4
  normCdf $(0, 10^{99}, 5, 9.22) \approx 0.706$ .
Answer
$\approx 0.706$ .
4Mean of $n$ iid copies
Stretch• sample mean
Question
$X_1, \ldots, X_4$ are iid with $E = 50, \text{Var} = 16$ . Let $\bar{X} = (X_1 + \cdots + X_4)/4$ . Find $E(\bar X)$ and $\text{Var}(\bar X)$ .
Step-by-step solution
1. Step 1
  $E(\bar X) = (1/4)(4 \cdot 50) = 50$ .
2. Step 2
  $\text{Var}(\bar X) = (1/4)^2 \cdot 4 \cdot 16 = (1/16)(64) = 4$ .
Answer
$E = 50$ , $\text{Var} = 4$ .

Key Definitions and Keywords — Further Probability

Definitions to memorise and the exact keywords mark schemes credit for further probability answers — sharpened from recent examiner reports for the 2026 IB DP Maths AA HL sitting.

Linearity of expectation
Examiner keyword
$E(aX + bY) = aE(X) + bE(Y)$ — holds always, no independence required.
Variance of linear transformation
Examiner keyword
$\text{Var}(aX + b) = a^2 \text{Var}(X)$ . Constant $b$ does not change variance.
Variance of independent sum
Examiner keyword
$\text{Var}(X \pm Y) = \text{Var}(X) + \text{Var}(Y)$ when $X, Y$ independent. Always ADD.

Common Mistakes and Misconceptions — Further Probability

The traps other students keep falling into on further probability questions — taken from recent IB DP Maths AA HL examiner reports and mark schemes — and how to avoid them.

✕ $\text{Var}(X - Y) = \text{Var}(X) - \text{Var}(Y)$ .
Why it happens
Misapplying linearity to variance.
How to avoid it
Variance is always non-negative — subtracting could go negative. The formula is PLUS regardless of sign.
✕ $\text{Var}(3X) = 3\text{Var}(X)$ .
Why it happens
Treating variance like expectation.
How to avoid it
Variance scales by the SQUARE of the coefficient: $\text{Var}(3X) = 9\text{Var}(X)$ .
✕Using $\text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y)$ without checking independence.
Why it happens
Skipping the assumption check.
How to avoid it
Always state 'X and Y are independent' before adding variances. Without it, $\text{Cov}(X, Y)$ enters the formula.

Past paper style quiz

Get a report showing which sub-topics you've nailed and which ones still need work.

Further Probability — frequently asked questions

The things students keep getting wrong in this sub-topic, answered.

Further Probability

Short Study Notes in the form of Slides

Short Study Notes — Further Probability

Detailed Notes

What you’ll learn

How it’s examined

1Linear transformation of one variable

2Sum of two independent variables

3Difference of normals

4Mean of nnn iid copies

Linearity of expectation

Variance of linear transformation

Variance of independent sum

✕Var(X−Y)=Var(X)−Var(Y)\text{Var}(X - Y) = \text{Var}(X) - \text{Var}(Y)Var(X−Y)=Var(X)−Var(Y).

✕Var(3X)=3Var(X)\text{Var}(3X) = 3\text{Var}(X)Var(3X)=3Var(X).

✕Using Var(X+Y)=Var(X)+Var(Y)\text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y)Var(X+Y)=Var(X)+Var(Y) without checking independence.

Past paper style quiz

Further Probability — frequently asked questions

4Mean of $n$ iid copies

✕ $\text{Var}(X - Y) = \text{Var}(X) - \text{Var}(Y)$ .

✕ $\text{Var}(3X) = 3\text{Var}(X)$ .

✕Using $\text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y)$ without checking independence.