Probability made clear.
A probability compares favourable outcomes with all possible outcomes. Start by describing the experiment so that the outcomes you count really are equally likely.
Key concepts in Probability
Basic probability
P(E) = favourable outcomes / total outcomes, when all elementary outcomes are equally likely.
Complement
P(not E) = 1 − P(E). This is especially helpful for “at least one”.
Independent events
If one result does not change another’s probability, multiply their probabilities for both occurring.
Without replacement
Removing an object changes both the total and sometimes the favourable count for the next draw.
Two fair coins: equally likely outcomes
| Outcome | Heads |
|---|---|
| HH | 2 |
| HT | 1 |
| TH | 1 |
| TT | 0 |
A reliable solving method
- State the experiment and whether replacement occurs.
- List or count an equally likely sample space.
- Translate “and”, “or” and “at least one” carefully.
- Check the answer is between 0 and 1.
Step-by-step solved examples
A fair die is rolled. Find the probability of a number greater than 4.
- Possible outcomes: 1,2,3,4,5,6.
- Favourable outcomes: 5 and 6, so there are two.
- Probability = 2/6 = 1/3.
Two fair coins are tossed. Find the probability of at least one head.
- Equally likely outcomes are HH, HT, TH and TT.
- Only TT has no head, with probability 1/4.
- Required probability = 1−1/4 = 3/4.
A bag has 3 red and 2 blue balls. Two are drawn without replacement. Probability both red?
- First red probability = 3/5.
- After one red is removed, second red probability = 2/4.
- Multiply: (3/5)(2/4) = 3/10.
Probability: practice questions
Choose one answer before opening the explanation. Write down your reasoning as well as the option; a correct guess is different from a method you can repeat.
Fair die: probability of an even number?
Show answer and explanation — question 1
Answer C: 1/2
The favourable faces are 2,4,6: 3/6 = 1/2.
Two fair coins: probability of exactly one head?
Show answer and explanation — question 2
Answer B: 1/2
HT and TH are two of four equally likely outcomes.
P(E)=0.35. Find P(not E).
Show answer and explanation — question 3
Answer C: 0.65
1−0.35 = 0.65.
One ball from 4 green and 6 yellow: probability green?
Show answer and explanation — question 4
Answer A: 2/5
4 out of 10 = 2/5.
Two independent fair die rolls: probability both six?
Show answer and explanation — question 5
Answer D: 1/36
Multiply 1/6 by 1/6.
A bag has 2 red and 3 blue. After drawing a red without replacement, chance next is blue?
Show answer and explanation — question 6
Answer B: 3/4
Three blue balls remain among four balls.
Frequently asked questions
Can probability exceed 1?
No. A value above 1 means the counting or arithmetic is wrong.
Is every two-outcome experiment 50–50?
No. Two outcomes can have different probabilities; fairness must be stated or justified.
State the experiment and whether replacement occurs. Check the answer is between 0 and 1.