Probability – Language, Scale, Calculation, Expected Outcomes (C8 – Cambridge IGCSE 0580)
1. What the exam expects you to know (C8)
- C8.1 – Introduction to probability
- Probability scale from 0 (impossible) to 1 (certain).
- Ability to express probabilities as fractions, decimals, percentages or odds.
- Complementary rule: \(P(E') = 1 - P(E)\).
- C8.2 – Relative and expected frequencies
- Convert observed frequencies into an empirical (relative) probability.
- Use the formula \(\text{expected frequency}= n \times P(E)\) to predict how many times an event would occur in a given number of trials.
- C8.3 – Probability of combined events
- Combined‑event questions are *with replacement* only (unless the question explicitly states otherwise).
- Be able to use Venn diagrams, tree diagrams or sample‑space diagrams to solve:
- Mutually exclusive (disjoint) events
- Independent events
- Dependent events (only when the question states “without replacement”).
2. Probability Language – Quick Reference
| Term (syllabus) |
Symbol used |
Meaning |
| Experiment |
– |
A process that produces one result from a set of possible results. |
| Outcome (result) |
– |
A single possible result of an experiment. |
| Sample space |
\(S\) |
The set of all possible outcomes. |
| Event |
\(E, F,\dots\) |
A subset of the sample space; any collection of outcomes. |
| Complement of an event |
\(E'\) (or \(\overline{E}\)) |
All outcomes in \(S\) that are not in \(E\). |
| Mutually exclusive (disjoint) events |
\(E\cap F=\varnothing\) |
Events that cannot occur together. |
| Independent events |
\(P(E\cap F)=P(E)P(F)\) |
The occurrence of one does not affect the probability of the other. |
| Probability of an event |
\(P(E)\) |
Numerical measure of how likely \(E\) is to occur ( \(0\le P(E)\le1\) ). |
3. Probability Scale
Probabilities can be written in four equivalent forms. All lie on the same 0 – 1 scale.
| Form |
Symbol |
Range |
Example |
| Fraction |
\(\dfrac{a}{b}\) |
\(0\le\dfrac{a}{b}\le1\) |
\(\dfrac{1}{4}\) |
| Decimal |
d |
\(0\le d\le1\) |
0.25 |
| Percentage |
p % |
\(0\% \le p\% \le 100\%\) |
25 % |
| Odds |
a : b |
\(a,b\ge0\) (not both zero) |
1 : 3 (equivalent to \(\dfrac{1}{4}\)) |
4. Single‑Event Probability
4.1 Theoretical (classical) probability
Used when every outcome in the sample space is equally likely.
\[
P(E)=\frac{\text{Number of favourable outcomes}}{\text{Total number of outcomes}}
\]
4.2 Empirical (experimental) probability & relative frequency
Based on actual observations. The empirical probability is the same as the relative frequency of the event.
\[
P_{\text{emp}}(E)=\frac{\text{Number of times }E\text{ occurs}}{\text{Total number of trials}}
\]
4.3 Expected frequency (expected number of occurrences)
For a large number of trials \(n\), the expected frequency of event \(E\) is
\[
\text{Expected frequency}=n\times P(E)
\]
4.4 Worked example – From observed frequency to expected frequency
A class tosses a fair coin 40 times and records 22 heads.
- Empirical probability of heads:
\(P_{\text{emp}}(H)=\dfrac{22}{40}=0.55\) (55 %).
- What number of heads would you expect in 120 tosses?
\(\text{Expected heads}=120\times0.55=66\).
5. Combined Events (C8.3)
Combined‑event questions are *with replacement* unless the question explicitly says “without replacement”. The following diagrammatic tools are used in the exam:
5.1 Venn diagrams
Useful for events that may overlap (e.g., “draw a red card or a face card”).
Example: From a standard deck (52 cards) find the probability of drawing a red card **or** a face card (with replacement).
- Red cards: 26.
- Face cards: 12 (J, Q, K of each suit).
- Red face cards (intersection): 6.
\[
P(R\cup F)=\frac{26}{52}+\frac{12}{52}-\frac{6}{52}=\frac{32}{52}=\frac{8}{13}\approx0.615\;(61.5\%)
\]
5.2 Tree diagrams
Best for sequential events, especially when the question states “with replacement”.
Example: Two dice are rolled, find the probability of obtaining a total of 7.
- First die: 6 equally likely outcomes.
- Second die (replacement): again 6 outcomes.
Only 6 ordered pairs give a total of 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1).
\[
P(\text{total}=7)=\frac{6}{36}=\frac{1}{6}\approx0.167\;(16.7\%)
\]
5.3 Sample‑space diagrams (grid method)
Draw a grid of all possible ordered pairs. Count favourable cells and divide by total cells.
Same example as above yields a 6‑out‑36 grid, giving \(\frac{6}{36}=\frac{1}{6}\).
5.4 Dependent events (without replacement)
Only used when the question explicitly says “without replacement”. The multiplication rule becomes
\[
P(E\cap F)=P(E)\times P(F\mid E)
\]
6. Rules for Combining Probabilities
- Complementary rule: \(P(E')=1-P(E)\).
- Addition rule (mutually exclusive): If \(E\cap F=\varnothing\), then \(P(E\cup F)=P(E)+P(F)\).
- General addition rule: \(P(E\cup F)=P(E)+P(F)-P(E\cap F)\).
- Multiplication rule (independent events): \(P(E\cap F)=P(E)P(F)\) (used for “with replacement”).
- Multiplication rule (dependent events): \(P(E\cap F)=P(E)P(F\mid E)\).
7. Worked Examples
Example 1 – Theoretical probability (single event)
Find the probability of obtaining an even number when a fair die is rolled.
- Sample space: \(S=\{1,2,3,4,5,6\}\).
- Favourable outcomes: \(E=\{2,4,6\}\) (3 outcomes).
\[
P(E)=\frac{3}{6}=\frac12=0.5=50\%
\]
Example 2 – Empirical probability (already covered in 4.4)
Example 3 – Expected frequency (single event)
Using the probability from Example 1, how many even numbers are expected in 120 rolls?
\[
\text{Expected frequency}=120\times\frac12=60\text{ even numbers.}
\]
Example 4 – Dependent events (without replacement)
A bag contains 3 red and 2 blue beads. Two beads are drawn successively without replacement. Find the probability that both are red.
\[
P(R_1)=\frac{3}{5},\qquad
P(R_2\mid R_1)=\frac{2}{4}=\frac12
\]
\[
P(R_1\cap R_2)=\frac{3}{5}\times\frac12=\frac{3}{10}=0.30=30\%
\]
Example 5 – Independent events (two coin tosses)
Find the probability of “heads then tails”.
\[
P(H\text{ then }T)=P(H)\times P(T)=\frac12\times\frac12=\frac14=0.25=25\%
\]
Example 6 – Combined events with a Venn diagram
From a standard deck, find the probability of drawing a red card **or** a face card (with replacement).
\[
P(R\cup F)=\frac{26}{52}+\frac{12}{52}-\frac{6}{52}=\frac{8}{13}\approx0.615\;(61.5\%)
\]
Example 7 – Expected value of a game
A player pays £2 to play. A fair die is rolled:
- Roll a 6 → win £10 (net gain £8).
- Roll 1–5 → win nothing (net loss £2).
| Net gain \(x_i\) |
Probability \(p_i\) |
\(x_i p_i\) |
| £8 |
\(\dfrac16\) |
\(\dfrac{8}{6}=1.\overline{3}\) |
| ‑£2 |
\(\dfrac56\) |
\(-\dfrac{10}{6}=-1.\overline{6}\) |
\[
E(X)=\sum x_i p_i=\frac{8}{6}-\frac{10}{6}=-\frac{2}{6}=-\frac13\approx -£0.33
\]
The negative expected value means the player loses, on average, about 33 pence per play.
8. Summary Checklist (What to do in every probability question)
- Identify the sample space \(S\) and the event(s) required.
- Decide whether the question is asking for a theoretical or empirical probability.
- Theoretical → use \(\displaystyle P(E)=\frac{\text{favourable}}{\text{total}}\).
- Empirical → use \(\displaystyle P_{\text{emp}}(E)=\frac{\text{occurrences}}{\text{trials}}\) (relative frequency).
- Choose the appropriate scale (fraction, decimal, % or odds) as required by the question.
- Determine if events are:
- Mutually exclusive → use the addition rule.
- Independent (with replacement) → use the multiplication rule \(P(E)P(F)\).
- Dependent (without replacement) → use \(P(E)P(F\mid E)\).
- If the question asks for “how many times” or “expected number”, compute \(n\times P(E)\).
- For expected‑value questions:
- List every possible net outcome \(x_i\).
- Write its probability \(p_i\).
- Calculate \(x_i p_i\) and sum the products.
- Check:
- All probabilities lie between 0 and 1 (or 0 % and 100 %).
- The answer matches the requested form.
- The expected value is sensible in the context of the problem.