IB Math AA SL Topic 4.7: Discrete random variables | Notes & Flashcards | Aimnova

IB Math AA SL — Discrete random variables

Topic 4.7 of IB Mathematics: Analysis and Approaches covers Discrete random variables, which is part of Unit 4: Statistics & Probability. Students explore key concepts including Distributions & E(X). A strong understanding of discrete random variables is essential for IB Math AA SL exams and builds the foundation for connected topics across the syllabus.

Key concepts in Discrete random variables

Key Idea: A discrete random variable lists outcomes and their probabilities in a table. The IB tests two things on it: finding a missing probability (the column must add to 1) and the expected value — the long-run average. Almost always Paper 1, by hand.

🎲 The probability distribution

\sum P(X = x) = 1

$X$: the discrete random variable — it takes separate values
$P(X = x)$: the probability of each value; each is between 0 and 1, and they sum to 1

📐 The three things you'll be asked

E(X) = \sum x\,P(X = x)

$x$: each value the variable can take
$E(X)$: the mean — the long-run average value of X

✏️ IB-style worked examples

IB-style question — find k so the probabilities sum to 1

A discrete random variable X has P(X = x) = kx for x = 2, 4, 6, 8. Find the value of k, then state P(X = 6).

Step by step:

Add all four probabilities and set the total equal to 1.
$k(2 + 4 + 6 + 8) = 20k = 1$
Solve for k.
$k = \tfrac{1}{20} = 0.05$
Substitute back to get the probability asked for.
$P(X = 6) = 6k = 6(0.05) = 0.3$

Final answer:

k = 0.05; P(X = 6) = 0.3.

IB-style question — compute the expected value E(X)

The number of goals X a team scores in a match has this distribution: P(X = 0) = 0.2, P(X = 1) = 0.5, P(X = 2) = 0.2, P(X = 3) = 0.1. Find E(X), the expected number of goals.

Step by step:

Multiply each value by its probability.
$E(X) = 0(0.2) + 1(0.5) + 2(0.2) + 3(0.1)$
Add the terms.
$= 0 + 0.5 + 0.4 + 0.3 = 1.2$

Final answer:

E(X) = 1.2 goals (a mean can be a value X never actually takes).

IB-style question — is the game fair?

At a stall you pay $3 to spin a wheel. You win $10 with probability 0.2, otherwise you win nothing. Find the expected net gain per play and state whether the game is fair.

Step by step:

Use the NET gain: +$7 if you win ($10 − $3 stake), −$3 if you lose.
$E(X) = 7(0.2) + (-3)(0.8)$
Work it out.
$= 1.4 - 2.4 = -1.0$
Compare with 0. A fair game needs E(X) = 0.
$E(X) = -1.0 \ne 0$

Final answer:

Expected net gain = −$1.00 per play, so the game is not fair (it favours the stall).

Important: In a fair-game question X is the net gain, so subtract the cost to play. A $10 prize that cost $3 is a net win of $7, not $10 — and a loss is the stake itself as a negative. Then set E(X) = 0 for a fair game; a positive E(X) favours the player, a negative one favours the house.

Tap each card to reveal the answer.

Exam Tips

Probabilities sum to 1 — set Σ P(X = x) = 1 to find an unknown, then substitute back.
E(X) = Σ x·P(X = x): multiply each value by its probability, then add.
E(X) need not be a value X can take — it's the long-run average.
For a game, let X be the NET gain (subtract the stake); it is fair when E(X) = 0.
This topic is non-calculator (Paper 1) — show the Σ working, not just the answer.

IB Math AA SL — Discrete random variables

Key concepts in Discrete random variables

Key Idea: A discrete random variable lists outcomes and their probabilities in a table. The IB tests two things on it: finding a missing probability (the column must add to 1) and the expected value — the long-run average. Almost always Paper 1, by hand.

🎲 The probability distribution

\sum P(X = x) = 1

$X$: the discrete random variable — it takes separate values
$P(X = x)$: the probability of each value; each is between 0 and 1, and they sum to 1

📐 The three things you'll be asked

E(X) = \sum x\,P(X = x)

$x$: each value the variable can take
$E(X)$: the mean — the long-run average value of X

✏️ IB-style worked examples

IB-style question — find k so the probabilities sum to 1

A discrete random variable X has P(X = x) = kx for x = 2, 4, 6, 8. Find the value of k, then state P(X = 6).

Step by step:

Add all four probabilities and set the total equal to 1.
$k(2 + 4 + 6 + 8) = 20k = 1$
Solve for k.
$k = \tfrac{1}{20} = 0.05$
Substitute back to get the probability asked for.
$P(X = 6) = 6k = 6(0.05) = 0.3$

Final answer:

k = 0.05; P(X = 6) = 0.3.

IB-style question — compute the expected value E(X)

The number of goals X a team scores in a match has this distribution: P(X = 0) = 0.2, P(X = 1) = 0.5, P(X = 2) = 0.2, P(X = 3) = 0.1. Find E(X), the expected number of goals.

Step by step:

Multiply each value by its probability.
$E(X) = 0(0.2) + 1(0.5) + 2(0.2) + 3(0.1)$
Add the terms.
$= 0 + 0.5 + 0.4 + 0.3 = 1.2$

Final answer:

E(X) = 1.2 goals (a mean can be a value X never actually takes).

IB-style question — is the game fair?

At a stall you pay $3 to spin a wheel. You win $10 with probability 0.2, otherwise you win nothing. Find the expected net gain per play and state whether the game is fair.

Step by step:

Use the NET gain: +$7 if you win ($10 − $3 stake), −$3 if you lose.
$E(X) = 7(0.2) + (-3)(0.8)$
Work it out.
$= 1.4 - 2.4 = -1.0$
Compare with 0. A fair game needs E(X) = 0.
$E(X) = -1.0 \ne 0$

Final answer:

Expected net gain = −$1.00 per play, so the game is not fair (it favours the stall).

Important: In a fair-game question X is the net gain, so subtract the cost to play. A $10 prize that cost $3 is a net win of $7, not $10 — and a loss is the stake itself as a negative. Then set E(X) = 0 for a fair game; a positive E(X) favours the player, a negative one favours the house.

Tap each card to reveal the answer.

Exam Tips

Probabilities sum to 1 — set Σ P(X = x) = 1 to find an unknown, then substitute back.
E(X) = Σ x·P(X = x): multiply each value by its probability, then add.
E(X) need not be a value X can take — it's the long-run average.
For a game, let X be the NET gain (subtract the stake); it is fair when E(X) = 0.
This topic is non-calculator (Paper 1) — show the Σ working, not just the answer.

IB Math AA SL — Discrete random variables

Key concepts in Discrete random variables

🎲 The probability distribution

📐 The three things you'll be asked

✏️ IB-style worked examples

IB-style question — find k so the probabilities sum to 1

IB-style question — compute the expected value E(X)

IB-style question — is the game fair?

Exam Tips

What you'll learn in Topic 4.7

Study resources — 4.7 Discrete random variables

Distributions & E(X)

Ready to study Discrete random variables?

Ready to practice?

IB Math AA SL — Discrete random variables

Key concepts in Discrete random variables

🎲 The probability distribution

📐 The three things you'll be asked

✏️ IB-style worked examples

IB-style question — find k so the probabilities sum to 1

IB-style question — compute the expected value E(X)

IB-style question — is the game fair?

Exam Tips

What you'll learn in Topic 4.7

Study resources — 4.7 Discrete random variables

Distributions & E(X)

Ready to study Discrete random variables?

Ready to practice?