Formal notation for independent and conditional events; the multiplication law for independent events; the conditional probability formula; dependent events such as selection without replacement.

What this dot point is asking

Edexcel codes 3p.08 and 3p.09 require you to know and apply the formal notation for independent events and for conditional probability, the multiplication law for independent events, and the conditional probability formula. The headline application is selection without replacement, where the probabilities change after the first selection, making the events dependent.

Independent events and notation

For independent events the multiplication law applies directly:

Independence typically arises with replacement or with genuinely separate experiments (a coin and a dice). The notation $P(A \mid B)$, read "the probability of $A$ given $B$", is the formal way Edexcel writes conditional probability.

Conditional probability

Rearranged, this gives the general multiplication law $P(A \text{ and } B) = P(A) \times P(B \mid A)$, which works even when the events are dependent. The simplest place to see conditional probability is a two-way table: "given that the person is a woman" means you only look at the women's row, and divide within it.

Dependent events and selection without replacement

When an item is selected and not replaced, the totals change, so the second selection is dependent on the first. On a tree diagram, the second set of branches has different probabilities depending on what was picked first. For example, drawing two counters from a bag without replacement: if the first is red, there is one fewer red and one fewer counter overall for the second draw. You multiply along the branches using these adjusted probabilities, exactly as $P(A) \times P(B \mid A)$.

Reading conditional probabilities from a two-way table

A two-way table makes conditional probability concrete. To find $P(\text{pass} \mid \text{woman})$, you ignore the men entirely: take the number of women who passed and divide by the total number of women. To find the unconditional $P(\text{pass})$, you divide total passes by the grand total. Keeping clear which total is the denominator is the key skill.

Testing for independence

You can use probabilities from a table to test whether two events are independent. Events $A$ and $B$ are independent if $P(A \mid B) = P(A)$, or equivalently if $P(A \text{ and } B) = P(A) \times P(B)$. So compare the conditional probability with the unconditional one: if "the probability of passing given you are a woman" equals "the probability of passing" overall, gender and passing are independent; if they differ, the events are dependent (gender is associated with passing). This is a common higher-mark task, and it links conditional probability to the idea of association from the correlation topics.

Venn diagrams for conditional probability

A Venn diagram is often the clearest tool for conditional probability. Once the numbers in each region (only $A$, only $B$, both, neither) are filled in, $P(B \mid A)$ is read as "the number in both, divided by the total number in $A$". For example, if $A$ contains $30$ people in total and $12$ of them are also in $B$, then $P(B \mid A) = \frac{12}{30} = 0.4$. Drawing the Venn diagram first turns an abstract conditional probability into a simple counting problem, and the same diagram answers "and", "or" and "not" questions too.