Conditional probability
Conditional probability is the probability of event A given that event B has occurred, written P(A | B) = P(A ∩ B) / P(B). It adjusts the likelihood of A by restricting the sample space to B.
See it move
Thirty per cent of a retailer's customers hold a loyalty card, and 18% both hold a card and made a purchase last month. Restricting attention to card holders — the conditioning event — gives P(purchase given card) = 18% ÷ 30% = 60%, well above the unconditional purchase rate. If that conditional probability had equalled the plain 30%, card ownership and purchasing would be independent.
The formula
Variables
- Probability of event A given that event B has occurred
- Probability that both A and B occur
- Probability of event B; must be greater than zero
Events A and B are statistically independent if P(A | B) = P(A), meaning knowing B gives no information about A.
Check yourself
A firm has 500 sales leads on its database. Of these, 150 are classified as 'warm' (prior contact made). Records show that 90 of the 150 warm leads converted into sales last quarter. What is the probability that a lead converted into a sale, given that it is classified as warm?