IB Math AI HL - Discrete Random Variables | Free Notes

What X means

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Probability distributions

What can happen — and how likely is it?: Toss a fair coin once. X counts the heads.

Result	X: number of heads	Probability
Tails	0
Heads	1

Read P(X = 1) as a question: What is the chance of getting 1 head?

Heads is 1 of the 2 equally likely results, so P(X = 1) = 1/2.

A probability distribution is a table like this: it lists every possible number X can be and its chance of happening.

Each chance is from 0 (impossible) to 1 (certain). The ∑ sign means add: all the chances together must equal 1.


0	0.20
1	0.35
2
3	0.15

Find a missing probability

Find k in the table.

Step by step

Set the total probability equal to 1.
Add the known probabilities and subtract from 1.

Final answer

k = 0.30, which is between 0 and 1.

Find a probability from the table: X ≥ 2 means X is 2 or 3 here.

Add their chances: 0.30 + 0.15 = 0.45.

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When probabilities come from a formula

Substitute each value of x: Put each possible number into the formula.

For x = 1, replace x with 1: (1 + 1) ÷ 9 = 2/9.

Only x = 1, 2 and 3 are possible.

	Substitute
1
2
3

Check the distribution

Is this a valid probability distribution?

Step by step

Each probability is between 0 and 1.
Check that they add to 1.

Final answer

Yes: both probability rules hold.

If the formula contains k: Find each probability in terms of k, then set their sum equal to 1.

Find k in a formula

P(X=x)=kx for x=1,2,3. Find k.

Step by step

Add all three chances.
Solve.

Final answer

k = 1/6.

Exam-style question

X has the probability distribution shown in the table. k is an unknown probability.

x	0	1	2
P(X = x)	0.2	k	0.3

Try it, then check

(a) Find k. [1]

(b) Find P(X ≥ 1). [2]

Step by step

All chances add to 1.
X ≥ 1 means 1 or 2.

Final answer

k = 0.5; P(X ≥ 1) = 0.8.

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Build a distribution from 3 coin tosses

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Two dice: keep the larger number

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A coin and a die

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The same method works for wheels: Grey: keep the number. Gold: add the bonus. Example: a score of 3 can come from grey + 3 or gold + 1 when the bonus is 2. Count the matching pairs, then divide by the total number of pairs.

Exam-style question

Try it, then check

Toss two fair coins. X counts heads. Build its distribution. [3]

Step by step

List all equally likely outcomes: TT, HT, TH, HH.
Group by heads: 0 from TT; 1 from HT or TH; 2 from HH.

Final answer

P(X=0)=1/4, P(X=1)=1/2, P(X=2)=1/4. Their sum is 1.

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Expected value

Multiply each value by its probability, then add.


0	0.1	0
1	0.3	0.3
2	0.4	0.8
3	0.2	0.6

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Long-run average, not one result: E(X) = 1.7 means the average approaches 1.7 over many tries.

One try still gives a whole score: 0, 1, 2 or 3.

Expected value on the GDC

A game has the probability distribution below. Calculate the expected score E(X) using your GDC. [2]

x	0	2	5	20
P(X = x)	0.5	0.3	0.15	0.05

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Show your method: For a simple table, also know how to write the expected-value substitution by hand.

For the GDC example: E(X) = 0(0.5) + 2(0.3) + 5(0.15) + 20(0.05) = 2.35.

Separate question: find the unknown value

X has the probability distribution below. Given that E(X) = 1, find n. [3]

x	−2	1	n
P(X = x)	0.4	0.45	0.15

IB-style questionFind[3 marks]

Given E(X) = 1, find n.

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Exam-style question

Try it, then check

A score is 0, 2 or 5 with probabilities 0.5, 0.3 and 0.2. Find E(X) and explain its meaning. [3]

Step by step

Multiply each score by its chance, then add.

Final answer

Over many plays, the average score approaches 1.6. One play gives 0, 2 or 5, never 1.6.

Prize, cost and gain

Gain includes the cost of playing: Gain = prize − cost to play.

Win $8 and pay $3: your gain is $5.

Win nothing and pay $3: your gain is −$3 (a loss).

A game is fair when the player’s expected net gain is zero.

Type 1: is it fair?

A player gains 3 dollars with probability 0.3 and loses 2 dollars with probability 0.7. Is the game fair?

Step by step

Use negative values for losses.
Interpret the expected gain.

Final answer

No. In the long run, the player loses an average of 0.50 dollars per game.

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Find a fair fee or gain

Find a gain that makes the game fair

Lose 8 points with probability 0.5; gain 4 points with probability 0.4; gain k points with probability 0.1.

For a fair game, :

, so .

What should the player pay so the game is fair?

Step by step

The four equally likely results pay $0, $4, $4 and $8. Add the prizes and divide by 4.
On average you receive $4. Pay $4 to leave no average gain or loss.

Final answer

Fair entry fee: $4.

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Revenue and compensation

Money, costs and compensation

Use the same method: identify numerical outcomes and their probabilities, multiply, add, then interpret.

A late delivery costs a shop 12 dollars compensation with probability 0.15; otherwise the cost is zero.

.

The long-run average compensation is 1.80 dollars per delivery.

Revenue after compensation; comparing options

For each outcome, subtract compensation from revenue first. Then calculate expected net revenue.

To compare two options, calculate the expected value for each and compare in context.

Average compensation

A shop pays €12 if a delivery is late. The chance is 0.15. Otherwise it pays €0.

Step by step

Multiply each cost by its chance.

Final answer

Average compensation is €1.80 per delivery.

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Exam-style question

Try it, then check

A game pays €0 with probability 0.75 or €8 with probability 0.25. It costs €3 to play. Is it fair? Find a fair fee. [4]

Step by step

Find the average prize.
Subtract the fee.

Final answer

Not fair: lose €1 on average. A fair fee is €2.

Use the average to make a decision

Option	What you receive
A: guaranteed	€50 every time
B: gamble	30% chance of €200; otherwise €0

Which would you choose?

Compare the guaranteed €50 with the gamble.

Step by step

Find the gamble’s average prize.
Choose B for the higher average (€60). Or choose A for a guaranteed €50 in one play.

Final answer

Either choice can be reasonable. Give the option and a reason: higher average or lower risk.

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Your current score can affect the gain

Result	Change in points: Y	Chance
Win 4 points	+4	0.5
Lose your current x points	−x	0.5

When does playing lose points on average?

You have x points. Y is the change in your score. Find when E(Y) is negative.

Step by step

Multiply each change by its chance and add.
For example, with 6 points, the average change is 2 − 3 = −1 point.
Find when the average change falls below zero.

Final answer

With more than 4 points, playing loses points on average. At exactly 4 points, the average change is zero.

Exam-style question

Try it, then check

You have x points. You gain 6 with probability 0.5, or lose all x with probability 0.5. Find E(Y), then when it is negative. [4]

Step by step

Y is the change: +6 or −x.
Set the average change below zero.

Final answer

Continuing loses points on average when x > 6.

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Two independent plays

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Score
0	0.2
1	0.5
2	0.3

Remember both orders: 0 then 2 and 2 then 0 are different possibilities.

Count 1 then 1 only once.

Expected number of results

Number of trials × probability of the event.

Expected losses

The probability of a loss is 0.25. A game is played 60 times. Find the expected number of losses.

Step by step

Multiply the number of games by the probability of a loss.

Final answer

15 losses are expected; exactly 15 losses are not guaranteed.

Asked for…	Use…
Average score or gain
Number of wins or losses

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Exam-style question

Try it, then check

Each play pays €0 or €4, each with probability 1/2. Find the chance of €4 total in two independent plays. In 40 plays, how many zero prizes are expected? [4]

Step by step

Two ways: 0 then 4, or 4 then 0.
Expected count = plays × chance.

Final answer

Chance of €4 total = 1/2. Expected zero prizes = 20.

Expected value (mean) and variance

Worked example

From coin example: find E(X) and Var(X).

Solution

E(X)=0(1/8)+1(3/8)+2(3/8)+3(1/8)
E(X)=0+3/8+6/8+3/8=12/8=1.5
E(X²)=0²(1/8)+1²(3/8)+2²(3/8)+3²(1/8)=24/8=3
Var(X)=3-1.5²=3-2.25=0.75

Final answer

E(X)=1.5, Var(X)=0.75, SD(X)=√0.75≈0.87.

IB-style question — find a value from E(X) [5 marks]

In a board game, a player moves a token and the score X (in points) has the distribution below, where n is a positive whole number.

x: −2, 1, n

P(X = x): 0.4, 0.45, 0.15

The expected score is E(X) = 1.

(a) Show that the probabilities given are consistent with a valid distribution.

(b) Find the value of n.

Step by step

(a) Check the probabilities add to 1.
They sum to 1 and each lies between 0 and 1, so the distribution is valid.
(b) Write the expected value with n still unknown and set it equal to 1.
Simplify the known terms.
Collect and solve for n.

Final answer

(a) The probabilities add to 1 (and each is between 0 and 1), so the distribution is valid. (b) n = 9.

IB-style question — is the game fair?

A game costs $2 to play. You win $5 (net gain $3) with probability 0.3, otherwise you lose your $2.

(a) Find the expected gain per play. (b) Is the game fair? (c) Find the expected number of wins in 50 plays.

Step by step

(a) E(gain) = Σ (gain × probability).
(b) A fair game has E = 0. Here E = −$0.50, so it is NOT fair (it favours the house).
(c) Expected number of wins = n × P(win) (this is np, not E(gain)).

Final answer

(a) −$0.50. (b) not fair. (c) 15 wins.

Multiply each gain by its probability and add for E(gain); a fair game needs that total to be 0. The expected COUNT of wins is a separate n×p calculation.

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Linear transformations

Key insight: E changes linearly.

Variance: only a² matters (not b).

Worked example

E(X)=1.5, Var(X)=0.75.

Find E(2X+5) and Var(2X+5).

Solution

E(2X+5)=2E(X)+5=2(1.5)+5=8
Var(2X+5)=2²Var(X)=4(0.75)=3

Final answer

E=8, Var=3, SD≈1.73.

Expected value when a new scoring formula is given

Use the formula, then average: An HL exploration can define a new score. You do not need to have memorised it: calculate each possible score, then multiply by its probability.

A surprise score

A sensor gives outputs A and B with probabilities 0.25 and 0.75. A question defines the surprise score S = −log₂(p), where p is the probability of the output obtained. Find E(S), and compare it with a sensor whose two outputs are equally likely.

Step by step

Calculate each score first.
Average the scores using their matching probabilities.
For two equally likely outputs both scores are 1.

Final answer

The equally likely sensor has the larger expected score: 1 rather than about 0.811. This supplied scoring rule is an example of applying expectation, not an extra formula to memorise.

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Compare income when penalties depend on the count

A fixed penalty is not a penalty per event: If a penalty is paid only after a threshold is crossed, multiply that penalty by the probability of crossing the threshold.

Two delivery offers

Three independent deliveries each have a 0.1 chance of being late. Offer A pays €45, minus €20 for each late delivery. Offer B pays €48, minus a single €10 penalty if at least two deliveries are late. Compare expected income.

Step by step

Let X count late deliveries; first find its expected count.
Offer A charges per late delivery.
Offer B charges only once when X ≥ 2.
Average that one-off penalty.

Final answer

Offer B gives the larger expected income: €47.72 compared with €39. This compares averages, not a guaranteed outcome.

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What X means

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Probability distributions

What can happen — and how likely is it?: Toss a fair coin once. X counts the heads.

Result	X: number of heads	Probability
Tails	0
Heads	1

Read P(X = 1) as a question: What is the chance of getting 1 head?

Heads is 1 of the 2 equally likely results, so P(X = 1) = 1/2.

A probability distribution is a table like this: it lists every possible number X can be and its chance of happening.

Each chance is from 0 (impossible) to 1 (certain). The ∑ sign means add: all the chances together must equal 1.


0	0.20
1	0.35
2
3	0.15

Find a missing probability

Find k in the table.

Step by step

Set the total probability equal to 1.
Add the known probabilities and subtract from 1.

Final answer

k = 0.30, which is between 0 and 1.

Find a probability from the table: X ≥ 2 means X is 2 or 3 here.

Add their chances: 0.30 + 0.15 = 0.45.

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When probabilities come from a formula

Substitute each value of x: Put each possible number into the formula.

For x = 1, replace x with 1: (1 + 1) ÷ 9 = 2/9.

Only x = 1, 2 and 3 are possible.

	Substitute
1
2
3

Check the distribution

Is this a valid probability distribution?

Step by step

Each probability is between 0 and 1.
Check that they add to 1.

Final answer

Yes: both probability rules hold.

If the formula contains k: Find each probability in terms of k, then set their sum equal to 1.

Find k in a formula

P(X=x)=kx for x=1,2,3. Find k.

Step by step

Add all three chances.
Solve.

Final answer

k = 1/6.

Exam-style question

X has the probability distribution shown in the table. k is an unknown probability.

x	0	1	2
P(X = x)	0.2	k	0.3

Try it, then check

(a) Find k. [1]

(b) Find P(X ≥ 1). [2]

Step by step

All chances add to 1.
X ≥ 1 means 1 or 2.

Final answer

k = 0.5; P(X ≥ 1) = 0.8.

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Build a distribution from 3 coin tosses

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Two dice: keep the larger number

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A coin and a die

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The same method works for wheels: Grey: keep the number. Gold: add the bonus. Example: a score of 3 can come from grey + 3 or gold + 1 when the bonus is 2. Count the matching pairs, then divide by the total number of pairs.

Exam-style question

Try it, then check

Toss two fair coins. X counts heads. Build its distribution. [3]

Step by step

List all equally likely outcomes: TT, HT, TH, HH.
Group by heads: 0 from TT; 1 from HT or TH; 2 from HH.

Final answer

P(X=0)=1/4, P(X=1)=1/2, P(X=2)=1/4. Their sum is 1.

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Expected value

Multiply each value by its probability, then add.


0	0.1	0
1	0.3	0.3
2	0.4	0.8
3	0.2	0.6

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Long-run average, not one result: E(X) = 1.7 means the average approaches 1.7 over many tries.

One try still gives a whole score: 0, 1, 2 or 3.

Expected value on the GDC

A game has the probability distribution below. Calculate the expected score E(X) using your GDC. [2]

x	0	2	5	20
P(X = x)	0.5	0.3	0.15	0.05

GDC walkthrough

Step through the exact calculator keystrokes, screen by screen, in study mode.

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Show your method: For a simple table, also know how to write the expected-value substitution by hand.

For the GDC example: E(X) = 0(0.5) + 2(0.3) + 5(0.15) + 20(0.05) = 2.35.

Separate question: find the unknown value

X has the probability distribution below. Given that E(X) = 1, find n. [3]

x	−2	1	n
P(X = x)	0.4	0.45	0.15

IB-style questionFind[3 marks]

Given E(X) = 1, find n.

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Exam-style question

Try it, then check

A score is 0, 2 or 5 with probabilities 0.5, 0.3 and 0.2. Find E(X) and explain its meaning. [3]

Step by step

Multiply each score by its chance, then add.

Final answer

Over many plays, the average score approaches 1.6. One play gives 0, 2 or 5, never 1.6.

Prize, cost and gain

Gain includes the cost of playing: Gain = prize − cost to play.

Win $8 and pay $3: your gain is $5.

Win nothing and pay $3: your gain is −$3 (a loss).

A game is fair when the player’s expected net gain is zero.

Type 1: is it fair?

A player gains 3 dollars with probability 0.3 and loses 2 dollars with probability 0.7. Is the game fair?

Step by step

Use negative values for losses.
Interpret the expected gain.

Final answer

No. In the long run, the player loses an average of 0.50 dollars per game.

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Find a fair fee or gain

Find a gain that makes the game fair

Lose 8 points with probability 0.5; gain 4 points with probability 0.4; gain k points with probability 0.1.

For a fair game, :

, so .

What should the player pay so the game is fair?

Step by step

The four equally likely results pay $0, $4, $4 and $8. Add the prizes and divide by 4.
On average you receive $4. Pay $4 to leave no average gain or loss.

Final answer

Fair entry fee: $4.

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Revenue and compensation

Money, costs and compensation

Revenue after compensation; comparing options

For each outcome, subtract compensation from revenue first. Then calculate expected net revenue.

To compare two options, calculate the expected value for each and compare in context.

Average compensation

A shop pays €12 if a delivery is late. The chance is 0.15. Otherwise it pays €0.

Step by step

Multiply each cost by its chance.

Final answer

Average compensation is €1.80 per delivery.

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Exam-style question

Try it, then check

A game pays €0 with probability 0.75 or €8 with probability 0.25. It costs €3 to play. Is it fair? Find a fair fee. [4]

Step by step

Find the average prize.
Subtract the fee.

Final answer

Not fair: lose €1 on average. A fair fee is €2.

Use the average to make a decision

Option	What you receive
A: guaranteed	€50 every time
B: gamble	30% chance of €200; otherwise €0

Which would you choose?

Compare the guaranteed €50 with the gamble.

Step by step

Find the gamble’s average prize.
Choose B for the higher average (€60). Or choose A for a guaranteed €50 in one play.

Final answer

Either choice can be reasonable. Give the option and a reason: higher average or lower risk.

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Your current score can affect the gain

Result	Change in points: Y	Chance
Win 4 points	+4	0.5
Lose your current x points	−x	0.5

When does playing lose points on average?

You have x points. Y is the change in your score. Find when E(Y) is negative.

Step by step

Multiply each change by its chance and add.
For example, with 6 points, the average change is 2 − 3 = −1 point.
Find when the average change falls below zero.

Final answer

With more than 4 points, playing loses points on average. At exactly 4 points, the average change is zero.

Exam-style question

Try it, then check

You have x points. You gain 6 with probability 0.5, or lose all x with probability 0.5. Find E(Y), then when it is negative. [4]

Step by step

Y is the change: +6 or −x.
Set the average change below zero.

Final answer

Continuing loses points on average when x > 6.

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Two independent plays

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Score
0	0.2
1	0.5
2	0.3

Remember both orders: 0 then 2 and 2 then 0 are different possibilities.

Count 1 then 1 only once.

Expected number of results

Number of trials × probability of the event.

Expected losses

The probability of a loss is 0.25. A game is played 60 times. Find the expected number of losses.

Step by step

Multiply the number of games by the probability of a loss.

Final answer

15 losses are expected; exactly 15 losses are not guaranteed.

Asked for…	Use…
Average score or gain
Number of wins or losses

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Exam-style question

Try it, then check

Each play pays €0 or €4, each with probability 1/2. Find the chance of €4 total in two independent plays. In 40 plays, how many zero prizes are expected? [4]

Step by step

Two ways: 0 then 4, or 4 then 0.
Expected count = plays × chance.

Final answer

Chance of €4 total = 1/2. Expected zero prizes = 20.

Expected value (mean) and variance

Worked example

From coin example: find E(X) and Var(X).

Solution

E(X)=0(1/8)+1(3/8)+2(3/8)+3(1/8)
E(X)=0+3/8+6/8+3/8=12/8=1.5
E(X²)=0²(1/8)+1²(3/8)+2²(3/8)+3²(1/8)=24/8=3
Var(X)=3-1.5²=3-2.25=0.75

Final answer

E(X)=1.5, Var(X)=0.75, SD(X)=√0.75≈0.87.

IB-style question — find a value from E(X) [5 marks]

Step by step

(a) Check the probabilities add to 1.
They sum to 1 and each lies between 0 and 1, so the distribution is valid.
(b) Write the expected value with n still unknown and set it equal to 1.
Simplify the known terms.
Collect and solve for n.

Final answer

(a) The probabilities add to 1 (and each is between 0 and 1), so the distribution is valid. (b) n = 9.

IB-style question — is the game fair?

Step by step

(a) E(gain) = Σ (gain × probability).
(b) A fair game has E = 0. Here E = −$0.50, so it is NOT fair (it favours the house).
(c) Expected number of wins = n × P(win) (this is np, not E(gain)).

Final answer

(a) −$0.50. (b) not fair. (c) 15 wins.

Multiply each gain by its probability and add for E(gain); a fair game needs that total to be 0. The expected COUNT of wins is a separate n×p calculation.

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Linear transformations

Key insight: E changes linearly.

Variance: only a² matters (not b).

Worked example

E(X)=1.5, Var(X)=0.75.

Find E(2X+5) and Var(2X+5).

Solution

E(2X+5)=2E(X)+5=2(1.5)+5=8
Var(2X+5)=2²Var(X)=4(0.75)=3

Final answer

E=8, Var=3, SD≈1.73.

Expected value when a new scoring formula is given

Use the formula, then average: An HL exploration can define a new score. You do not need to have memorised it: calculate each possible score, then multiply by its probability.

A surprise score

Step by step

Calculate each score first.
Average the scores using their matching probabilities.
For two equally likely outputs both scores are 1.

Final answer

The equally likely sensor has the larger expected score: 1 rather than about 0.811. This supplied scoring rule is an example of applying expectation, not an extra formula to memorise.

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Compare income when penalties depend on the count

A fixed penalty is not a penalty per event: If a penalty is paid only after a threshold is crossed, multiply that penalty by the probability of crossing the threshold.

Two delivery offers

Step by step

Let X count late deliveries; first find its expected count.
Offer A charges per late delivery.
Offer B charges only once when X ≥ 2.
Average that one-off penalty.

Final answer

Offer B gives the larger expected income: €47.72 compared with €39. This compares averages, not a guaranteed outcome.

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Probability distributions

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