Ad Banner Placeholder

Cambridge IGCSE Mathematics — 0580 Extended

Topic 8.6: Probability — Conditional Probability

“Given B” means the sample space has shrunk to B. You no longer divide by \(n(\mathcal{E})\). On 0580 you do this from a Venn, a tree or a table — the papers do not require the notation \(\mathrm{P}(A \mid B)\) or a formula sheet identity. Read the word given, restrict, then count.

From a Venn

Given B, look only inside circle B. Favourable means the overlap \(A \cap B\). Denominator is \(n(B)\), not \(n(\mathcal{E})\).

Method

  1. Underline the given set. That circle (or row, or first branch) is the new whole.
  2. Count how many of those also match the other event.
  3. Write (both) \(\div\) (given group).

From the Venn with \(n(B) = 20\) and \(n(A \cap B) = 7\): a person is chosen at random from those in B. Find the probability they are also in A.

Venn with set B ringed as the new sample space; 7 of the 20 in B are also in A
7 out of the 20 in B, so \(7/20\). Do not divide by 40.

From a tree

If you are told the first event has happened, stay on that first branch. The second-stage fraction on that branch is already the conditional probability.

A bag has 3 red and 2 blue. Two counters are taken without replacement. Given the first was red, find the probability the second is blue.

Without-replacement tree with the red-first half emphasised; P(second blue given first red) = 2/4
The blue-first half of the tree is irrelevant once the first is known to be red. The answer is \(2/4 = 1/2\).

From a two-way table

The given row (or column) is the new sample space. Read the cell, then the row total.

80 students. 45 play football, 30 play tennis, 18 play both. A student is chosen at random from those who play football. Find the probability they also play tennis.

Two-way table of football and tennis with the football row highlighted: 18/45 = 2/5
Given football → use 45, not 80. Also tennis → 18, so \(18/45 = 2/5\).

Without replacement is already conditional

Each second-branch fraction is “given what has just been taken”. You do not need extra notation: redraw the bag, then write the new fraction.

Two bags of remaining counters: after red, 2 red and 2 blue; after blue, 3 red and 1 blue
After red, \(\mathrm{P}(\text{red next}) = 2/4\). After blue, it is \(3/4\).

Why is “A and B” different from “A given B”?

Two Venns: overlap 7 out of 40 versus overlap 7 out of the 20 in B
Same overlap, different denominator. \(\mathrm{P}(A\text{ and }B) = 7/40\). Given B, it is \(7/20\).

Paper 2 (non-calculator)

\(18/45 = 2/5\) by cancelling 9. The given total is the denominator — cancelling does not change that fact.

Try this

From the football/tennis table, a student is chosen from those who play tennis. Find the probability they play football.

Show answer
Answer

  1. Given tennis → column total 30. Football and tennis → 18

    \[ \dfrac{18}{30} = \dfrac{3}{5} \]

Exam Traps

  • The word given changes the denominator. Dividing by the whole of \(\mathcal{E}\) (40, or 80, or 5/5) is the mark lost on almost every conditional question.

0/10

Ad Banner Placeholder