What X means
Interactive diagram
Explore the labelled diagram, charts and maps for this topic in full study mode.
Free preview
This is the free notes preview
You're reading the free notes. Aimnova Pro unlocks the full study experience — and you can try it with your first topic free to keep:
- FlashcardsLock in vocabulary and key terms with spaced repetition.
- Practice questionsAnswer exam-style questions and get instant AI marking.
- Mock exams & past-paper vaultSit full mocks and see exactly how examiners award marks.
- Personalised study planA daily plan built around your exam date and weak areas.
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.
| 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.
Memorize terms 3x faster
Smart flashcards show you cards right before you forget them. Perfect for definitions and key concepts.
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.
| 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.