Conditional Probability

This lesson covers conditional probability at A-Level, extending the probability concepts from earlier lessons. Conditional probability is the probability of an event occurring given that another event has already occurred. It is essential for solving problems involving dependent events and for understanding Bayes' theorem.

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Definition of Conditional Probability

The conditional probability of event $A$A given event $B$B has occurred is:

$P(A | B) = \frac{P(A \cap B)}{P(B)} \quad \text{provided } P(B) > 0$P(A∣B)=P(B)P(A∩B)​provided P(B)>0

This can be rearranged to give the multiplication rule:

$P(A \cap B) = P(A | B) \times P(B)$P(A∩B)=P(A∣B)×P(B)

Example: In a group of 100 students, 60 study maths and 30 study both maths and physics. What is the probability a student studies physics given they study maths?

$P(\text{Physics} | \text{Maths}) = \frac{P(\text{Maths} \cap \text{Physics})}{P(\text{Maths})} = \frac{30/100}{60/100} = \frac{30}{60} = 0.5$P(Physics∣Maths)=P(Maths)P(Maths∩Physics)​=60/10030/100​=6030​=0.5

Exam Tip: The notation $P(A|B)$P(A∣B) reads as "the probability of $A$A given $B$B". The event after the vertical bar is the condition — it is what you know has happened. Always identify clearly which event is the condition.

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Two-Way Tables and Conditional Probability