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C8.1 Introduction to probability

Probability measures how likely an event is on a scale from 0 to 1 inclusive. A larger number means the event is more likely; every valid probability lies between the two endpoints.

Probability	Meaning
$0$	impossible
between $0$ and $0.5$	unlikely
$0.5$	equally likely to occur or not occur
between $0.5$ and $1$	likely
$1$	certain

For a fair six-sided spinner numbered 1 to 6, landing on 4 has probability $\tfrac16$ , so its arrow belongs one sixth of the way from 0 to 1. Landing on an even number has probability $\tfrac36=0.5$ , while landing on 9 has probability 0 because 9 is not an outcome.

Convert a fraction to a decimal when the scale is marked in decimal intervals. For example, $\tfrac35=0.6$ , so place it just to the right of the midpoint. Compare probabilities by their values, not by the size of the event's label.

A probability cannot be negative or greater than 1. The scale describes likelihood, not the number of times an event must occur in a short experiment; observed frequency is treated in C8.2.

When all possible outcomes are equally likely, the probability of one event is the number of favourable outcomes divided by the total number of possible outcomes.

\text{probability}=\frac{\text{number of favourable outcomes}}{\text{total number of equally likely outcomes}}

List or count the whole outcome set once, count only outcomes that satisfy the event, form the fraction, then simplify or convert to a decimal or percentage if requested. Information may come from a list, frequency table, graph or a Venn diagram with at most two sets.

A fair seven-sided spinner is numbered 1 to 7. The prime outcomes are $2,3,5,7$ , so there are 4 favourable outcomes out of 7 and the probability is $\tfrac47$ . For red and green balls in the ratio $2:3$ , the probability of red is $\tfrac{2}{2+3}=\tfrac25$ .

The favourable-over-total rule needs equally likely individual outcomes. Do not divide by only the favourable count, and do not assume that categories are equally likely when a table or spinner gives unequal weights. Your final fraction, decimal or percentage must represent the same value.

An event and its 'not' event are complements: exactly one of them must occur, so their probabilities add to 1.

\text{probability of not }A=1-\text{probability of }A

Given	Complement calculation	Result
fraction $\tfrac7{10}$	$1-\tfrac7{10}$	$\tfrac3{10}$
decimal $0.47$	$1-0.47$	$0.53$
percentage $18\%$	$100\%-18\%$	$82\%$

With counts, subtract the event count from the total before dividing. In a table, graph or two-set Venn diagram, first identify every region belonging to the event; everything else is its complement. This avoids missing outcomes outside a named set.

If the probability that a train is late is $0.15$ , the probability that it is not late is $1-0.15=0.85$ . The check is $0.15+0.85=1$ .

Not $A$ means every allowed outcome outside $A$ , not an unrelated event you choose. Subtract from 1, not from 100 unless you are working in percentages, and keep the result between 0 and 1.

C8.2 Relative and expected frequencies

Relative frequency is the proportion of trials in which an event occurs. It gives an experimental estimate of the event's probability.

\text{relative frequency}=\frac{\text{number of times the event occurs}}{\text{total number of trials}}

A spinner is used 80 times and lands on 5 on 13 occasions. The relative frequency of landing on 5 is $\tfrac{13}{80}=0.1625$ , so $0.1625$ is an estimate of the probability of landing on 5.

A different set of trials can give a different relative frequency. A larger, well-run sample usually gives a more stable estimate because one unusual result has less influence, but it does not guarantee the exact theoretical probability.

To combine experiments, add the event counts and add the trial counts, then divide the two totals. For example, results of 8 successes from 30 trials and 17 from 70 trials combine to $\tfrac{8+17}{30+70}=0.25$ . Do not take an unweighted average when the trial totals differ.

Relative frequency is evidence from observed results, not a certainty about the next result. 'Random' means an individual result is unpredictable; it does not mean that every outcome must have equal probability.

Expected frequency is the number of occurrences predicted from a probability over a stated number of trials or members of a population.

\text{expected frequency}=\text{probability of the event}\times\text{number of trials or population size}

A bag contains 7 red, 5 green and 2 pink counters. A counter is selected at random, replaced and the experiment is repeated 140 times. Since $P(\text{green})=\tfrac5{14}$ , the expected frequency of green is $\tfrac5{14}\times140=50$ . Replacement keeps the probability the same for each trial.

If the probability is unknown, estimate it with relative frequency first. In a representative sample, 36 of 240 people have a feature, so the estimated probability is $\tfrac{36}{240}=0.15$ . For a population of 1600, the expected number is $0.15\times1600=240$ .

Term	Meaning
fair	the relevant outcomes have equal probabilities
biased	the mechanism favours some relevant outcomes, so their probabilities are not equal
random	an individual result cannot be predicted with certainty; probabilities need not be equal

An expected frequency is a long-run prediction, not a guarantee of the actual count. Keep the probability unrounded until the final multiplication. A calculation may give a non-integer expectation even though an observed count must be a whole number; round only when the context or question requires it.

C8.3 Probability of combined events

A combined event joins two or more conditions. Choose a representation that lists every possible outcome once, then combine only the outcomes that satisfy the event.

Representation	Use it when	How probability is found
sample space diagram	two stages can be arranged as rows and columns	mark the favourable cells, then divide by all equally likely cells
two-set Venn diagram	outcomes are classified by membership of $A$ and $B$	add the required region counts once, then divide by the universal total
tree diagram	events happen in stages	multiply probabilities along each required path; add the probabilities of alternative paths

P(\text{one path})=P(\text{first branch})\times P(\text{second branch}),\qquad P(\text{alternative paths})=\text{sum of their path probabilities}

A bag contains 6 red, 3 green and 1 blue marble. Two marbles are selected with replacement. The probability of green stays $\tfrac3{10}$ on the second stage, so $P(\text{green then green})=\tfrac3{10}\times\tfrac3{10}=\tfrac9{100}$ . Write outcomes at the ends of branches and probabilities beside the branches.

For two fair six-sided dice, a $6\times6$ sample space has 36 equally likely ordered pairs. A total of 5 occurs at $(1,4),(2,3),(3,2),(4,1)$ , so its probability is $\tfrac4{36}=\tfrac19$ . Ordered pairs such as $(1,4)$ and $(4,1)$ are different outcomes.

In a Venn diagram, $A\text{ or }B$ includes both single-set regions and the overlap, but the overlap is counted once. $A\text{ and }B$ is only the overlap. This syllabus uses at most two sets, and combined-event tree problems here use replacement, so branch probabilities do not change after the first selection.

8. Probability

Exam analysis

Most tested topics

C8.1 Introduction to probability

Read and use the probability scale

Calculate the probability of one event

Find the probability that an event does not occur

C8.2 Relative and expected frequencies

Use relative frequency to estimate probability

Calculate expected frequency and interpret fair, biased and random

C8.3 Probability of combined events

Calculate probabilities of combined events