Probability measures how likely an event is to happen. It is always a number from 0 (impossible) to 1 (certain). Events in between are described as unlikely, even chance, or likely.
Example 1 — Describing probability in words
Step 1
A bag has 3 red and 7 blue counters. One is picked at random.
Step 2
\(P(\text{red}) = \frac{3}{10} = 0.3\) — closer to 0 than to 0.5, so unlikely
Step 3
\(P(\text{blue}) = \frac{7}{10} = 0.7\) — closer to 1 than to 0.5, so likely
Example 2 — Placing a value on the scale
Step 1
A spinner has 5 equal sections numbered 1–5. Find \(P(\text{number} > 1)\).
Step 2
Numbers greater than 1: 2, 3, 4, 5 — four outcomes out of 5.
Step 3
\(P(\text{number} > 1) = \frac{4}{5} = 0.8\) — place near the likely/certain end.
Key vocabulary: Impossible (0) · Unlikely (close to 0) · Even chance (0.5) · Likely (close to 1) · Certain (1). Probability is never negative and never greater than 1.
✓ Quick Check 1 — The Probability Scale
Question 1 of 10
Probability as Fraction, Decimal and Percentage
Probability can be written in three equivalent forms. The key rule to know is the complement rule: if you know \(P(\text{event})\), you can always find \(P(\text{not event})\) by subtracting from 1.
When we cannot calculate a theoretical probability, we estimate it from experiments. The relative frequency is the proportion of trials in which the event occurred. More trials means a more reliable estimate.
\[\text{Relative frequency} = \frac{\text{number of successful outcomes}}{\text{total number of trials}}\]
Example 5 — Calculating relative frequency
Step 1
A drawing pin is dropped 100 times. It lands point up 35 times.
Trial B is a more reliable estimate — more trials reduce the effect of random variation.
Relative frequency vs theoretical probability: Use relative frequency when you have experimental data. Use the theoretical formula (favourable ÷ total equally-likely outcomes) for fair dice, coins, and cards.
✓ Quick Check 3 — Relative Frequency
Question 1 of 10
Practice: Relative Frequency Calculator
Enter the number of successes and total trials, then click Calculate.
Practice: The Probability Scale
Answer each question, then click Check All.
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Question
Answer
Result
1
What is P(a certain event)?
2
What is P(an impossible event)?
3
Convert the probability 3/4 to a decimal
4
Convert the probability 0.2 to a percentage
5
P(rain) = 0.35. What is P(no rain)?
6
Coin flipped 200 times, 92 heads. Relative frequency of heads?
7
Spinner spun 50 times, lands on red 20 times. Relative frequency of red?
8
P(win) = 0.15. What is P(lose)?
9
Convert the probability 0.6 to a fraction (simplest form)
10
Die rolled 120 times, lands on 6 exactly 20 times. Relative frequency of a 6?
⚠ Things to Watch Out For
Using numbers outside 0 to 1: Probability is ALWAYS between 0 and 1 inclusive. A probability of 1.5 or −0.2 is impossible — recheck your calculation.
P(event) + P(not event) = 1: If P(rain) = 0.3, then P(no rain) = 0.7. These must sum to 1. This is one of the most useful probability rules.
Confusing "unlikely" with "impossible": Unlikely means low probability (but not zero). Impossible means probability = 0 (it cannot happen at all). These are different.
Relative frequency vs theoretical probability: Relative frequency (from experiments) approaches theoretical probability only with large numbers of trials. Fewer trials give less reliable estimates.
Expressing probability as a percentage or fraction: Probability can be expressed as a fraction, decimal, or percentage — they are equivalent. 0.4 = 2/5 = 40%. Make sure you convert correctly.