What Is Probability?#
Probability measures how likely an event is to happen, on a scale from 0 (impossible) to 1 (certain).
| Value | Meaning | Example |
|---|---|---|
| $P = 0$ | Impossible | Rolling a 7 on a standard die |
| $0 < P < 0.5$ | Unlikely | Rolling a 6 on a standard die ($\frac{1}{6} \approx 0.17$) |
| $P = 0.5$ | Even chance | Flipping heads on a fair coin |
| $0.5 < P < 1$ | Likely | Drawing a red card from a full deck ($\frac{26}{52} = 0.5$… actually even!) |
| $P = 1$ | Certain | Rolling a number less than 7 on a standard die |
Rule: All probability answers must satisfy $0 \leq P(A) \leq 1$. If your answer is negative or greater than 1, you’ve made an error.
1. Theoretical Probability#
$$\boxed{P(\text{Event}) = \frac{n(E)}{n(S)} = \frac{\text{Number of favourable outcomes}}{\text{Total number of possible outcomes}}}$$- $S$ = the sample space (set of ALL possible outcomes)
- $E$ = the event (the outcomes you want)
- $n(S)$ = size of the sample space
- $n(E)$ = number of favourable outcomes
Worked Example 1 — Die#
A fair die is rolled. Find $P(\text{even number})$.
$S = \{1;\; 2;\; 3;\; 4;\; 5;\; 6\}$, so $n(S) = 6$
Even numbers: $E = \{2;\; 4;\; 6\}$, so $n(E) = 3$
$$P(\text{even}) = \frac{3}{6} = \frac{1}{2} = 0.5 = 50\%$$Worked Example 2 — Cards#
A card is drawn from a standard deck of 52 cards. Find $P(\text{heart})$ and $P(\text{face card})$.
Hearts: 13 hearts in a deck. $P(\text{heart}) = \frac{13}{52} = \frac{1}{4}$
Face cards (Jack, Queen, King): 4 suits × 3 = 12 face cards.
$P(\text{face card}) = \frac{12}{52} = \frac{3}{13}$
Worked Example 3 — Balls in a Bag#
A bag contains 5 red, 3 blue, and 2 green balls. One ball is drawn at random.
$n(S) = 5 + 3 + 2 = 10$
$P(\text{red}) = \frac{5}{10} = \frac{1}{2}$
$P(\text{blue or green}) = \frac{3 + 2}{10} = \frac{5}{10} = \frac{1}{2}$
$P(\text{not blue}) = \frac{5 + 2}{10} = \frac{7}{10}$
2. The Complement Rule#
The complement of event A (written $A'$ or “not A”) is everything that is NOT in A.
$$\boxed{P(A') = 1 - P(A)}$$This works because either A happens or it doesn’t — there’s no third option.
Worked Example 4#
If $P(\text{rain}) = 0.3$, find $P(\text{no rain})$.
$P(\text{no rain}) = 1 - 0.3 = 0.7$
Worked Example 5#
A bag has 8 red and 12 blue balls. Find $P(\text{not red})$.
$P(\text{red}) = \frac{8}{20} = \frac{2}{5}$
$P(\text{not red}) = 1 - \frac{2}{5} = \frac{3}{5}$
Check: “Not red” = blue = $\frac{12}{20} = \frac{3}{5}$ ✓
The Power of the Complement: “At Least One”#
Whenever a problem asks for “at least one”, it’s almost always easier to calculate:
$$P(\text{at least one}) = 1 - P(\text{none})$$This avoids listing every possible “at least one” case.
3. Listing Sample Spaces#
For combined experiments, you need to list outcomes systematically to avoid missing any.
Two Coins#
$S = \{HH;\; HT;\; TH;\; TT\}$, $n(S) = 4$
$P(\text{exactly one head}) = \frac{2}{4} = \frac{1}{2}$ (the outcomes HT and TH)
$P(\text{at least one head}) = 1 - P(\text{no heads}) = 1 - \frac{1}{4} = \frac{3}{4}$
A Coin and a Die#
$n(S) = 2 \times 6 = 12$ outcomes:
$S = \{H1;\; H2;\; H3;\; H4;\; H5;\; H6;\; T1;\; T2;\; T3;\; T4;\; T5;\; T6\}$
$P(\text{Heads and even}) = \frac{3}{12} = \frac{1}{4}$ (the outcomes H2, H4, H6)
Two Dice — The Full Sample Space#
Two dice give $6 \times 6 = 36$ outcomes. It’s essential to use a grid to see them all:
| 1 | 2 | 3 | 4 | 5 | 6 | |
|---|---|---|---|---|---|---|
| 1 | (1,1) | (1,2) | (1,3) | (1,4) | (1,5) | (1,6) |
| 2 | (2,1) | (2,2) | (2,3) | (2,4) | (2,5) | (2,6) |
| 3 | (3,1) | (3,2) | (3,3) | (3,4) | (3,5) | (3,6) |
| 4 | (4,1) | (4,2) | (4,3) | (4,4) | (4,5) | (4,6) |
| 5 | (5,1) | (5,2) | (5,3) | (5,4) | (5,5) | (5,6) |
| 6 | (6,1) | (6,2) | (6,3) | (6,4) | (6,5) | (6,6) |
Important: (3, 5) and (5, 3) are different outcomes — the first die shows 3 and the second shows 5, or vice versa.
Worked Example 6 — Two Dice#
Two dice are rolled. Find $P(\text{sum} = 7)$.
Count the outcomes where the sum is 7: $(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)$ → 6 outcomes
$$P(\text{sum} = 7) = \frac{6}{36} = \frac{1}{6}$$Worked Example 7 — Two Dice#
Find $P(\text{double})$ (both dice show the same number).
Doubles: $(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)$ → 6 outcomes
$$P(\text{double}) = \frac{6}{36} = \frac{1}{6}$$Worked Example 8 — Two Dice#
Find $P(\text{sum} > 9)$.
Outcomes with sum > 9 (sum = 10, 11, or 12):
Sum 10: $(4,6), (5,5), (6,4)$ → 3
Sum 11: $(5,6), (6,5)$ → 2
Sum 12: $(6,6)$ → 1
Total: 6 outcomes
$$P(\text{sum} > 9) = \frac{6}{36} = \frac{1}{6}$$4. Relative Frequency (Experimental Probability)#
Relative frequency is probability estimated from actual experiments:
$$\text{Relative Frequency} = \frac{\text{Number of times event occurred}}{\text{Total number of trials}}$$Worked Example 9#
A coin is flipped 200 times and lands on heads 118 times.
Relative frequency of heads = $\frac{118}{200} = 0.59$
Theoretical probability = $0.5$
The relative frequency (0.59) is close to but not exactly 0.5. This is normal for a small number of trials.
The Law of Large Numbers#
The more trials you perform, the closer the relative frequency gets to the theoretical probability. This is the Law of Large Numbers.
- 10 flips might give 7 heads (70%) — very different from 50%
- 100 flips might give 55 heads (55%) — closer
- 10 000 flips might give 5 023 heads (50.23%) — very close to 50%
Worked Example 10#
A spinner has sections coloured red, blue, and green. After 500 spins: red appears 210 times, blue 175 times, green 115 times. Estimate the probability of each colour.
$P(\text{red}) \approx \frac{210}{500} = 0.42$
$P(\text{blue}) \approx \frac{175}{500} = 0.35$
$P(\text{green}) \approx \frac{115}{500} = 0.23$
Check: $0.42 + 0.35 + 0.23 = 1.00$ ✓
Notice: The probabilities are NOT equal, suggesting the spinner sections are NOT equal in size.
5. Expressing Probability — Three Forms#
Probability can be expressed as a fraction, decimal, or percentage:
$$P = \frac{1}{4} = 0.25 = 25\%$$| Form | When to use |
|---|---|
| Fraction | When the answer is exact (e.g., $\frac{1}{6}$) |
| Decimal | When comparing probabilities or doing calculations |
| Percentage | When communicating to a general audience |
Worked Example 11#
Express $P(\text{sum} = 7) = \frac{6}{36}$ in all three forms.
$\frac{6}{36} = \frac{1}{6} \approx 0.167 \approx 16.7\%$
🚨 Common Mistakes#
| Mistake | Why it’s wrong | Fix |
|---|---|---|
| Probability > 1 | $P$ must be between 0 and 1. Check your denominator | $n(E)$ can never exceed $n(S)$ |
| Not listing the full sample space | Two dice have $6 \times 6 = 36$ outcomes, not 12 or 6 | Draw the grid — it shows every outcome |
| $(3,5)$ = $(5,3)$ | These are different outcomes — die 1 shows 3 and die 2 shows 5, or the reverse | Always treat the dice as distinguishable |
| Confusing $P(A')$ with $P(A)$ | “Not red” means everything EXCEPT red (blue AND green) | Use $P(A') = 1 - P(A)$ |
| Forgetting to simplify | $\frac{4}{12}$ should be $\frac{1}{3}$ | Always simplify fractions |
| Confusing theoretical and experimental probability | Theoretical uses the formula; experimental uses actual trial data | State which type you’re calculating |
| Not checking that all probabilities sum to 1 | The probabilities of all possible outcomes must add to exactly 1 | Add them up as a check |
💡 Pro Tips for Exams#
1. The Complement Shortcut#
Whenever you see “at least one,” use:
$P(\text{at least one}) = 1 - P(\text{none})$
This is almost always easier than listing all the “at least one” cases.
2. The Two-Dice Grid#
For any two-dice question, draw the 6×6 grid. It takes 30 seconds and prevents counting errors. Circle the outcomes you want, count them, divide by 36.
3. The “Check” Habit#
After calculating a probability:
- Is it between 0 and 1? ✓
- Does it make intuitive sense? ✓
- If you calculated multiple probabilities, do they sum to 1? ✓
🔗 Related Grade 10 topics:
- Venn Diagrams — visualising probability with the addition rule and intersections
- Statistics — relative frequency connects probability to data analysis
📌 Where this leads in Grade 11:
- Venn Diagrams & Logic — the addition rule and testing for independence
- Independent & Dependent Events — tree diagrams, contingency tables, conditional probability
🏠 Back to Probability | ⏭️ Venn Diagrams