Probability
Grade 9 · Math · Study Guide
Calculate theoretical and experimental probability, work with compound events, and apply counting principles.
Probability measures how likely an event is to occur, expressed as a number between 0 (impossible) and 1 (certain). The theoretical probability of an event is P(event) = (number of favorable outcomes) / (total number of equally likely outcomes). For example, flipping a fair coin gives P(heads) = 1/2.
Compound events involve two or more events. For independent events (one doesn't affect the other), the multiplication rule gives P(A and B) = P(A) × P(B). For mutually exclusive events (can't both happen), the addition rule gives P(A or B) = P(A) + P(B). If events can overlap, use P(A or B) = P(A) + P(B) − P(A and B) to avoid double-counting.
Experimental probability is based on actual trials: P(event) = (number of times event occurred) / (total number of trials). As the number of trials increases, experimental probability approaches theoretical probability — this is called the Law of Large Numbers.
| Rule | Formula | When to Use |
|---|---|---|
| Complement rule | P(not A) = 1 − P(A) | When it is easier to find the opposite event |
| Multiplication (independent) | P(A and B) = P(A) × P(B) | Two independent events both occur |
| Addition (mutually exclusive) | P(A or B) = P(A) + P(B) | Either event occurs, but not both |
| Addition (overlapping) | P(A or B) = P(A) + P(B) − P(A and B) | Events can occur at the same time |
Key terms
- Probability
- A measure from 0 to 1 of how likely an event is to occur
- Theoretical probability
- P(event) = favorable outcomes / total outcomes (assuming equally likely)
- Experimental probability
- Probability based on observed results from actual trials
- Independent events
- Events where the outcome of one does not affect the other
- Mutually exclusive events
- Events that cannot both occur at the same time
- Complement
- All outcomes NOT in an event; P(not A) = 1 − P(A)
Flashcards
- Probability
- P(event) = favorable outcomes / total outcomes; value from 0 to 1
- Complement rule
- P(not A) = 1 − P(A)
- Independent events
- P(A and B) = P(A) × P(B)
- Mutually exclusive
- Events that cannot occur at the same time; P(A and B) = 0
- Experimental probability
- Observed occurrences ÷ total trials from an actual experiment
- Law of Large Numbers
- As trials increase, experimental probability approaches theoretical probability
Practice questions
A bag has 3 red and 7 blue marbles. What is P(red)?
- 3/10
- 7/10
- 3/7
- 1/3
Show answer
Answer: 3/10
Total marbles = 10; favorable = 3 red; P = 3/10.
The probability of an event can be greater than 1.
Show answer
Answer: False
Probability is always between 0 (impossible) and 1 (certain), inclusive.
If P(A) = 0.4, then P(not A) = ___.
Show answer
Answer: 0.6
P(not A) = 1 − 0.4 = 0.6.
You roll a fair die twice. What is P(rolling a 3 both times)?
- 1/36
- 1/6
- 2/6
- 1/12
Show answer
Answer: 1/36
P(3) × P(3) = 1/6 × 1/6 = 1/36 (independent events).
Events A and B are mutually exclusive. P(A) = 0.3 and P(B) = 0.4. What is P(A or B)?
- 0.12
- 0.7
- 0.1
- 1.0
Show answer
Answer: 0.7
Mutually exclusive: P(A or B) = 0.3 + 0.4 = 0.7.
Fun facts
- Casinos are built on probability — every game is designed so the house wins slightly more than 50% of the time, which adds up to billions.
- The probability of being struck by lightning in a given year is about 1 in 1,222,000 — but over 80 years, it rises to about 1 in 15,000.
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