Probability: language, scale, calculation, expected outcomes

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.

1. Empirical probability of heads:
   \(P_{\text{emp}}(H)=\dfrac{22}{40}=0.55\) (55 %).
2. 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.

1. First die: 6 equally likely outcomes.
2. 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)

1. Identify the sample space \(S\) and the event(s) required.
2. 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).
3. Choose the appropriate scale (fraction, decimal, % or odds) as required by the question.
4. 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 \(P(E)P(F\mid E)\).
5. If the question asks for “how many times” or “expected number”, compute \(n\times P(E)\).
6. 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.
7. 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.