Probability by reasoning
Theoretical probability is calculated without doing an experiment. When all outcomes are equally likely:
P(event) = number of favourable outcomes ÷ total number of possible outcomes
A favourable outcome is one that counts as the event happening.
Example: a die
The sample space for a fair die is {1, 2, 3, 4, 5, 6}.
| Event | Favourable outcomes | Probability |
|---|---|---|
| Roll a 5 | {5} | 1/6 |
| Roll an even number | {2, 4, 6} | 3/6 = 1/2 |
| Roll more than 4 | {5, 6} | 2/6 = 1/3 |
| Roll a 7 | none | 0 |
| Roll less than 7 | all | 1 |
The probability scale
Probabilities go from 0 (impossible) to 1 (certain). They can be fractions, decimals or percents: 1/4 = 0.25 = 25%.
Example: a bag of marbles
A bag has 5 red, 3 blue and 2 green marbles (10 total).
- P(red) = 5/10 = 1/2
- P(blue) = 3/10
- P(not green) = 8/10 = 4/5
The complement
The complement of an event is everything else. Since something must happen:
P(not A) = 1 − P(A)
- P(not rolling a 6) = 1 − 1/6 = 5/6
A deck of cards
A standard deck has 52 cards: 4 suits (hearts, diamonds, clubs, spades) of 13 cards each (A, 2–10, J, Q, K).
| Event | Probability |
|---|---|
| Draw a heart | 13/52 = 1/4 |
| Draw a king | 4/52 = 1/13 |
| Draw a red card | 26/52 = 1/2 |
| Draw a face card (J, Q, K) | 12/52 = 3/13 |
Using theoretical probability to predict
Expected number = P(event) × number of trials.
Roll a die 300 times: expected number of 3s = 1/6 × 300 = 50.
Only for equally likely outcomes
The formula only works if outcomes are equally likely. A spinner with one large red section and a small blue section isn’t 1/2 red: you’d use the angles instead.
Worked examples
Example 1: P(spinning a 2) on a spinner with 8 equal sections numbered 1–8?
1/8.
Example 2: The letters of MATHEMATICS are put in a bag. P(drawing an M)?
2 M’s out of 11 letters: 2/11.
Example 3: P(not drawing a red card) from a standard deck?
1 − 1/2 = 1/2.
Common mistakes
- Using the formula when outcomes aren’t equally likely.
- Counting the sample space wrong (forgetting duplicates, like two M’s).
- Writing probabilities greater than 1.
Try it yourself
- P(rolling a number less than 3)?
- A bag has 4 yellow and 6 purple counters. P(yellow)?
- P(drawing an ace) from a standard deck?
- P(not rolling a 1)?
- Expected heads in 80 coin flips?
Answers: 1) 1/3 2) 2/5 3) 1/13 4) 5/6 5) 40
Quick recap
- Theoretical probability = favourable ÷ total equally likely outcomes.
- List the sample space carefully.
- P(not A) = 1 − P(A).