Conditional Probability [H]

Conditional Probability

The conditional probability \( P(B|A) \) is the probability of event \( B \) occurring given that event \( A \) has already occurred. The condition restricts the sample space to outcomes in which \( A \) has happened.

The Formula

\[ P(B|A) = \frac{P(A\cap B)}{P(A)} \]

Equivalently: \( P(A\cap B)=P(A)\times P(B|A) \) — the general multiplication rule.

Calculating Conditional Probability from Tables

The most reliable GCSE method is to work with expected frequencies (counts) from a two-way table rather than using the formula directly. Restrict attention to the row or column defined by the given condition, then read off the count for the required event.

Three Tools

  • Two-way table: row/column restriction — most common at GCSE.
  • Tree diagram: read off the conditional probability directly from the second-stage branch after the conditioning event.
  • Venn diagram: the condition restricts to one circle; count within that circle.

Worked Examples

120 students: 70 study Maths (M), 80 study Science (S), 50 study both. One student is chosen at random. Find \( P(S|M) \).

Students studying Maths: 70. Of those, 50 also study Science.

\[ P(S|M) = \frac{50}{70} = \frac{5}{7} \]
A tree diagram has \( P(A)=0.4 \), and given A the probability of B is 0.7. Find \( P(A\cap B) \) and \( P(B|A) \).
\[ P(A\cap B) = 0.4\times0.7 = 0.28 \]

\( P(B|A)=0.7 \) — read directly from the second branch after A.

Venn diagram: \( n(A)=30 \), \( n(B)=25 \), \( n(A\cap B)=10 \), total=60. Find \( P(A|B) \).

Given B (25 students): 10 are also in A.

\[ P(A|B) = \frac{10}{25} = \frac{2}{5} \]

 Key Takeaways

  • \( P(B|A) \) = probability of B given A has occurred — restrict the sample space to A.
  • Formula: \( P(B|A)=P(A\cap B)/P(A) \). But working with counts from tables is usually simpler.
  • General multiplication rule: \( P(A\cap B)=P(A)\times P(B|A) \).
  • Independent events: \( P(B|A)=P(B) \) — knowing A happened gives no information about B.
  • Conditional probability always lies between 0 and 1.