Probability
What is Probability?
Probability measures how likely an event is to happen. It ranges from 0 (impossible) to 1 (certain).
P(event) = Number of favourable outcomes / Total number of possible outcomes
Probability can be expressed as a fraction (1/6), a decimal (0.1667), or a percentage (16.67%). All three are equally valid.
Key Terms
- Outcome: a single result of a probability experiment
- Event: one or more outcomes we are interested in
- Sample space: the complete set of all possible outcomes
- Favourable outcome: an outcome that satisfies the event we are measuring
- Mutually exclusive: events that cannot occur at the same time
- Independent events: events where the outcome of one does not affect the other
The Probability Scale
0 โโโโโโโโโโ 0.5 โโโโโโโโโโ 1
impossible even chance certain
- P = 0: the event cannot happen (e.g. rolling a 7 on a standard die)
- P = 0.5: the event is equally likely to happen or not (e.g. flipping heads)
- P = 1: the event will definitely happen (e.g. rolling a number less than 7)
Calculating Probability
A bag has 3 red, 5 blue, and 2 green balls (10 total):
| Event | Calculation | Probability |
|---|---|---|
| P(blue) | 5/10 | 1/2 |
| P(red) | 3/10 | 0.3 |
| P(green) | 2/10 | 1/5 |
| P(not red) | 7/10 | 0.7 |
Complementary Events
Every event has a complement โ the event not happening.
P(event) + P(not event) = 1
So: P(not event) = 1 - P(event)
Examples:
- If P(rain tomorrow) = 0.3, then P(no rain) = 1 - 0.3 = 0.7
- If P(winning) = 1/5, then P(not winning) = 1 - 1/5 = 4/5
Experimental vs Theoretical Probability
Theoretical probability is calculated from equally likely outcomes using the formula above. It tells us what should happen in the long run.
Experimental probability is based on actual results of repeated trials: P(event) = number of times event occurred / total number of trials
Example: A coin is flipped 100 times and lands heads 43 times. Experimental P(heads) = 43/100 = 0.43
The theoretical P(heads) = 0.5
As the number of trials increases, experimental probability gets closer to theoretical probability. This is called the Law of Large Numbers.
Mutually Exclusive Events
Two events are mutually exclusive if they cannot both occur at the same time.
For mutually exclusive events: P(A or B) = P(A) + P(B)
Example: On a single die roll, rolling a 2 and rolling a 5 are mutually exclusive. P(2 or 5) = 1/6 + 1/6 = 2/6 = 1/3
Warning: this addition rule only works for mutually exclusive events.
Independent Events โ Multiplying Probabilities
Two events are independent if the outcome of one does not affect the other.
For independent events: P(A and B) = P(A) ร P(B)
Example: Flipping a coin twice. P(heads on flip 1) = 1/2 P(heads on flip 2) = 1/2 P(two heads) = 1/2 ร 1/2 = 1/4
Tree Diagrams
Tree diagrams display all possible outcomes of a sequence of events. You multiply probabilities along branches and add probabilities of separate paths.
Example: Flip a coin, then roll a die.
- Branch 1: Heads (P = 1/2) โ Roll 4 (P = 1/6) โ P(H and 4) = 1/12
- Branch 2: Tails (P = 1/2) โ Roll 4 (P = 1/6) โ P(T and 4) = 1/12
The probabilities of all final branches must sum to 1.
Common Mistakes to Avoid
- Adding when you should multiply: for independent events happening together, multiply probabilities โ do not add them
- Using the addition rule for non-mutually-exclusive events: if A and B can both happen, the rule P(A or B) = P(A) + P(B) overcounts
- Expressing probability greater than 1: probability is always between 0 and 1 (or 0% and 100%)
- Confusing experimental and theoretical: a coin landing heads 3 times in a row does not mean the next flip is more likely to be tails
Tips and Tricks
- Always check that all probabilities in a sample space add up to 1
- Use tree diagrams when there are two or more sequential events
- For "at least one" problems, it is often easier to use: P(at least one) = 1 - P(none)
- Label branches clearly with both the outcome and its probability