Topic 4.7: exam essentials
| Remember | Meaning |
|---|---|
| $\sum P(X=x)=1$ | All probabilities add to 1 |
| $E(X)=\sum xP(X=x)$ | Multiply each value by its probability; add |
| Expected value | Long-run average |
| $E(\text{gain})=0$ | Fair game |
| Gain = prize − cost | Money left after paying to play |
| $np$ | Expected number of occurrences |
| Independent trials | Multiply within each pair |
| Several possible pairs | Add their probabilities |
Which method do I use? Missing chance: subtract the known chances from 1. Average score: multiply each score by its chance, then add. Fair game: average gain must be 0. How many losses: games × chance of losing. Two independent plays: list the pairs, multiply their chances, then add. Formula: put each allowed value into it.
Choose the right skill
Discrete random variables & probability distributions
| Exam skill | Method |
|---|---|
| Missing probability | Known chances + missing chance = 1 |
| Probability formula | Substitute each allowed x |
Building probability distributions
| Exam skill | Method |
|---|---|
| Build a distribution | List → group by X → count → divide |
Expected value
| Exam skill | Method |
|---|---|
| Expected value | Multiply each value by its chance, then add |
| Interpret | Average over many trials; may not occur in one trial |
Fair games & expected money
| Exam skill | Method |
|---|---|
| Fair game | Average net gain = 0 |
| Fair fee | Fee = average prize |
Expected value in decisions
| Exam skill | Method |
|---|---|
| Choose an option | Compare averages and risk; explain your choice |
| Outcome depends on x | Include −x for losing your current points |
Repeated trials & expected outcomes
| Exam skill | Method |
|---|---|
| Two independent trials | Multiply within pairs; add different pairs |
| Expected count | Number of trials × chance |
Combined exam-style question
This original question combines building a distribution, expected gain, a fair fee, expected losses and two independent plays.
A game uses two fair coins. The prize is 0 dollars for no heads, 4 dollars for one head and 8 dollars for two heads. A player pays 3 dollars to play. (a) Find the probability distribution of the player’s net gain X. [3] (b) Find E(X) and interpret your answer. [2] (c) Find the entry fee that makes the game fair. [2] (d) The game is played 60 times. Find the expected number of losses. [2] (e) Two plays are independent. Find the probability that the total prize is 8 dollars. [3]
(a) What do you gain or lose?
Key Idea: X = prize − 3 dollars. A negative X means you lose money.
| Coin result | Prize | X: prize − cost |
|---|---|---|
| TT: no heads | 0 dollars | 0 − 3 = −3 dollars |
| HT: one head | 4 dollars | 4 − 3 = 1 dollar |
| TH: one head | 4 dollars | 4 − 3 = 1 dollar |
| HH: two heads | 8 dollars | 8 − 3 = 5 dollars |
4 equally likely results: each has probability 1/4. HT and TH both give X = 1, so add their chances: 1/4 + 1/4 = 1/2.
| X: gain or loss | Results giving this X | Probability |
|---|---|---|
| −3 dollars | TT | $1/4$ |
| 1 dollar | HT or TH | $1/2$ |
| 5 dollars | HH | $1/4$ |
(b) Average gain
Find E(X) and explain what it means.
Step by step:
(b) Multiply each net gain by its probability and add.
Interpret: the player gains an average of 1 dollar per game in the long run.
Over many games, the player gains an average of 1 dollar per game.
(c) A fair entry fee
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.
Fair entry fee: $4.
Interactive diagram
Explore the labelled diagram, charts and maps for this topic in study mode.
(d) Expected losses
Using the original 3-dollar entry fee, how many losses do we expect in 60 games?
Step by step:
You lose money only with TT: you pay 3 dollars and win nothing. TT is 1 of the 4 equally likely results.
The chance of losing is 1/4. Take one quarter of 60 games.
Expect 15 losses on average. A particular set of 60 games could have more or fewer.
(e) Total prize from two plays
Find the chance that the two prizes add to 8 dollars.
Step by step:
There are three ways to make $8: $0 then $8, $4 then $4, or $8 then $0.
For each pair, multiply the two chances. The plays are independent.
Add the chances of these three different ways.
Probability = 3/8 = 0.375.
A prize is before the entry fee. A gain is after the fee. In part (e), the total is a prize total, so use prizes 0, 4 and 8.
Additional HL tools
Key Idea: A discrete random variable takes countable values, each with a probability. Its expected value is the long-run mean, and its variance measures how spread out the values are.
Distributions and E(X)
- The probabilities of a distribution add up to 1.
- E(X) = Σ x P(X=x): multiply each value by its probability and add.
- On the GDC: values in L1, probabilities in L2, then 1-Var Stats. The x̄ it shows is E(X).
- To judge a game, compare E(X) with the cost. A negative expected profit means a loss.
| Rule | Formula |
|---|---|
| Variance | Var(X)=E(X²)-[E(X)]² |
| Standard deviation | SD=√Var |
| Linear change | Var(aX+b)=a²Var(X) |
| Independent X, Y | Var(X+Y)=Var(X)+Var(Y) |
Adding a constant shifts the mean but leaves the spread alone.
P(X = x) is 0.5, 0.3, k, 0.05. Find k. k = 1 − 0.85 = 0.15.
Prizes are 0, 2, 5 and 20 with those probabilities. Find E(X). 2.35.
A spin costs 3. Should a player expect a profit? No. 2.35 − 3 = −0.65, a loss of 65 cents per spin.
Var(X) = 0.81. Find Var(3X − 2). 9 × 0.81 = 7.29.