Practice Maths

L33 — Probability Distributions

Key Terms

Random variable X
A variable whose value is a numerical outcome of a random experiment.
Probability distribution
A complete list of all possible values of X and their probabilities; all P ≥ 0 and they must sum to 1.
Expected value E(X)
The long-run average outcome; E(X) = ∑ x ⋅ P(X = x). Not necessarily a possible value of X.
Uniform distribution
All outcomes equally likely; P(X = x) = 1/n for n outcomes. E(X) = (a + b) / 2 for values a to b.
Mode
The value of X with the highest probability — the most likely single outcome.
Binomial expectation
For n independent trials each with P(success) = p: E(X) = np.

Discrete probability distributions

A discrete probability distribution lists all possible values of a discrete random variable X and their probabilities.

RequirementRule
All probabilities valid0 ≤ P(X = x) ≤ 1 for all x
Probabilities sum to 1∑ P(X = x) = 1

Expected value (mean)

E(X) = ∑ x ⋅ P(X = x)

The expected value is the long-run average outcome over many trials.

Uniform distribution

Each outcome equally likely: P(X = x) = 1/n for n outcomes. Example: rolling a fair die (n = 6).

Common distributions at Grade 10

DistributionDescriptionE(X)
UniformAll outcomes equally likely(a + b) / 2
Binomial-likeFixed trials, two outcomes (success/fail)np (n trials, P(success) = p)
0 0.1 0.2 0.3 0.4 1 2 3 4 5 x P(X=x) E(X) = 3
Probability distribution: symmetric, E(X) = 1(0.1)+2(0.2)+3(0.4)+4(0.2)+5(0.1) = 3
Hot Tip: E(X) is the long-run average, not a guaranteed outcome — it may not even be a possible value of X. For a fair die, E(X) = 3.5, but no single roll gives 3.5. E(X) is accurate as a prediction only over many repeated trials.

Worked Example 1 — Verifying a probability distribution

X can take values 1, 2, 3, 4 with P = 0.1, 0.3, k, 0.2. Find k and verify.

0.1 + 0.3 + k + 0.2 = 1 ⇒ k = 0.4.

Check: all values 0 ≤ P ≤ 1. Sum = 1. Valid distribution.

Worked Example 2 — Expected value

X takes values 1, 2, 3 with P = 1/4, 1/2, 1/4. Find E(X).

E(X) = 1(1/4) + 2(1/2) + 3(1/4) = 1/4 + 1 + 3/4 = 2.

Worked Example 3 — Game fairness

A game costs $2 to play. You roll a die: win $5 if you roll a 6, $0 otherwise. Is the game fair?

Winnings: $5 with P = 1/6, $0 with P = 5/6.

E(winnings) = 5(1/6) + 0(5/6) = 5/6 ≈ $0.83.

Net gain per game = $0.83 − $2 = −$1.17. The game favours the house — not fair to the player.

Worked Example 4 — Distribution from a table

A box has 2 red and 3 blue balls. Two drawn without replacement. Let X = number of red balls drawn. Find the distribution of X.

X can be 0, 1, or 2.

P(X=0) = (3/5)(2/4) = 6/20 = 3/10.

P(X=1) = (2/5)(3/4) + (3/5)(2/4) = 6/20 + 6/20 = 12/20 = 3/5.

P(X=2) = (2/5)(1/4) = 2/20 = 1/10.

Check: 3/10 + 6/10 + 1/10 = 10/10 = 1. ✓

E(X) = 0(3/10) + 1(6/10) + 2(1/10) = 0 + 6/10 + 2/10 = 8/10 = 4/5.

Worked Example 5 — Binomial expectation

A fair coin is flipped 10 times. Let X = number of heads. Find E(X).

n = 10, p = 0.5. E(X) = np = 10 × 0.5 = 5.

We expect 5 heads on average over many sets of 10 flips.

See Answers ➔

  1. Valid distributions. Fluency

    • (a) X takes values 1, 2, 3 with P = 0.2, 0.5, 0.3. Is this a valid distribution?
    • (b) X takes values 0, 1, 2 with P = 0.3, 0.4, k. Find k.
    • (c) X takes values 1, 2, 3, 4 with P = 1/8, 3/8, 3/8, k. Find k.
    • (d) Can P(X = 5) = −0.1 be part of a valid distribution? Explain.

  2. Expected value. Fluency

    • (a) X: values 0, 1, 2 with P = 0.4, 0.4, 0.2. Find E(X).
    • (b) X: values 1, 2, 3, 4 with P = 0.1, 0.3, 0.4, 0.2. Find E(X).
    • (c) A fair die is rolled. Find E(X) where X is the number shown.
    • (d) X: values −1, 0, 1 with P = 1/4, 1/2, 1/4. Find E(X).

  3. Game expectation. Fluency

    • (a) A lottery ticket costs $1. P(win $10) = 0.05, P(win $0) = 0.95. Find the expected profit per ticket.
    • (b) A raffle sells 500 tickets at $2 each. Prize is $400. Find E(profit) for a ticket buyer.
    • (c) A game: flip a coin. Win $3 for heads, lose $2 for tails. Find E(net winnings).
    • (d) What expected value makes a game exactly fair (no advantage to either side)?

  4. Distributions from scenarios. Fluency

    • (a) Two coins are flipped. Let X = number of heads. Write the distribution of X.
    • (b) A bag has 1 red and 4 blue balls. One ball is drawn. Let X = 1 if red, 0 if blue. Find P(X = 1).
    • (c) Die rolled once. X = score if even, X = 0 if odd. Write the distribution of X.
    • (d) In (c), find E(X).

  5. Probability distribution from a diagram. Understanding

    The bar chart shows the probability distribution of X, the number of pets owned by a randomly selected household.

    0 0.10 0.20 0.30 0.40 0 0.15 1 0.35 2 0.30 3 0.15 4 0.05 Number of pets (x) P(X=x)
    • (a) Verify that the probabilities sum to 1.
    • (b) Find P(X ≥ 2).
    • (c) Find E(X).
    • (d) What is the most likely number of pets? What is this value called?

  6. Distribution from counting. Understanding

    A box has 3 red and 2 white balls. Two balls are drawn without replacement. X = number of red balls drawn.

    • (a) List all possible values of X.
    • (b) Find P(X = 0), P(X = 1), P(X = 2).
    • (c) Verify the probabilities sum to 1 and find E(X).
    • (d) Interpret E(X) in context.

  7. Expected winnings. Understanding

    A prize wheel has 12 equal sections: 6 sections win $0, 3 sections win $5, 2 sections win $10, 1 section wins $50.

    • (a) Write the probability distribution of the prize X.
    • (b) Find E(X).
    • (c) If the wheel costs $4 to spin, find the expected profit/loss per spin.
    • (d) How many spins would you need to expect a total profit of $20 for the operator?

  8. Comparing distributions. Understanding

    Two investment options. Option A: gain $200 (P=0.5), gain $50 (P=0.3), lose $100 (P=0.2). Option B: gain $120 (P=0.7), lose $50 (P=0.3).

    • (a) Find E(gain) for Option A.
    • (b) Find E(gain) for Option B.
    • (c) Which option has the higher expected gain?
    • (d) Option A has higher variability (risk). If a risk-averse investor prefers stability, which should they choose?

  9. Designing a fair game. Problem Solving

    A carnival game involves rolling two dice. If the sum is 7 or 11, the player wins. The game costs $c to play and pays $10 on a win.

    • (a) Find P(sum = 7) and P(sum = 11).
    • (b) Find P(win) = P(sum = 7 or 11).
    • (c) Find E(net winnings) in terms of c.
    • (d) Find c to make the game exactly fair for the player.

  10. Insurance expected value. Problem Solving

    A home insurance policy costs $800/year. If there is a fire (P = 0.002), the insurer pays $150 000. If there is a minor incident (P = 0.05), they pay $3 000. Otherwise they pay nothing.

    • (a) Find the expected payout E(payout) per policy.
    • (b) Find the insurer’s expected profit per policy.
    • (c) The insurer has 10 000 policies. Find the expected total profit.
    • (d) Explain why insurance is still a rational purchase for homeowners despite negative expected value for them.