A table of outcomes and their probabilities: A discrete random variable X takes separate values, each with a probability. A probability distribution lists them in a table. The probabilities must be valid (each between 0 and 1) and sum to 1.
IB-style question — is it valid?
A variable X has P(X = 1) = 0.1, P(X = 2) = 0.3, P(X = 3) = 0.4, P(X = 4) = 0.2. Show that this is a valid probability distribution.
Step by step
- Each probability is between 0 and 1, and add them.
- Since they total 1, it is valid.
Final answer
All probabilities are valid and sum to 1, so it is a valid distribution.
The total is always 1: If the probabilities don't add to 1, either a value is missing or there's an error.
Use Σ P = 1 to solve for the unknown: If a probability (or a parameter inside one) is unknown, set the sum of all probabilities equal to 1 and solve.
IB-style question — find k
A distribution has P(X = x) = kx for x = 1, 2, 3, 4. Find k, then P(X = 3).
Step by step
- Sum the probabilities and set equal to 1.
- Solve, then substitute.
Final answer
k = 0.1; P(X = 3) = 0.3.
Substitute back: After finding the parameter, substitute to get the actual probability the question asks for.
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Sum of value × probability: The expected value (mean) of X is E(X) = Σ x·P(X = x) — multiply each value by its probability and add. It's the long-run average value of X.
IB-style question — find E(X)
X has P(X = 1) = 0.1, P(X = 2) = 0.2, P(X = 3) = 0.3, P(X = 4) = 0.4. Find E(X).
Step by step
- Multiply each value by its probability.
- Add.
Final answer
E(X) = 3.
E(X) need not be a possible value: The mean can be a value X never takes (e.g. 2.7) — it's an average, not an outcome.
E(X) = the long-run average gain: For a game, let X be the net gain. E(X) is the average gain per play. A game is fair when E(X) = 0 (no expected gain or loss). Set E(X) = 0 to find a fair prize.
IB-style question — make it fair
You roll a fair die. You win $w if it shows a six, otherwise you lose $2. Find the prize w that makes the game fair.
Step by step
- Net gain: +w with prob 1/6, −2 with prob 5/6. Set E(X) = 0.
- Solve for w.
Final answer
A prize of $10 makes the game fair.
Use the NET gain: Include the cost or loss in X. If a $5 prize costs $5 to enter, the net 'win' is $0, not $5.