Understand conditional probability and connect independence to unchanged conditional probabilities.
Find \(P(A \mid B)\) from counts by dividing within the conditioned group
Problem
Among the \(30\) students who play sports, \(12\) play soccer. Compute \(P(\text{soccer}~|~\text{sports})\) by dividing the overlap count by the conditioned-group count.
Big Picture
What this problem is really about
The word given restricts the denominator to the conditioned group. We’ll identify that reduced universe, count the favorable overlap inside it, form favorable-over-conditioned rather than using the overall total, reduce the fraction, and interpret the result within the conditioned population.
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