11Fconditional


 * Conditional Probability **


 * Pr(A | B) means the probability that A occurs __**given that**__ B has already happened.

math . \qquad \qquad \qquad Pr(A \, | \, B) = \dfrac{ Pr(A \, \cap \, B)}{Pr(B)} math


 * ** Example **
 * 50 students nominate which sport they would like to play
 * 30 choose Basketball (B)
 * 25 choose Soccer (S)
 * 5 choose neither of those
 * This information is shown in the Venn Diagram


 * What is the probability that a student wants to play Basketball __**given that**__ they want to play soccer?

math . \qquad \qquad \qquad Pr( Basketball \; | \; Soccer) \\. \\ . \qquad \qquad \qquad Pr(B \, | \, S) = \dfrac{ Pr(B \, \cap \, S) }{Pr(S)} math . . math . \qquad \qquad \qquad \qquad \qquad Pr(B \, \cap \, S) = \dfrac{10}{50} = \dfrac{1}{5} \\. \\ . \qquad \qquad \qquad \qquad \qquad Pr(S) = \dfrac{25}{50} = \dfrac{1}{2} math . . math . \qquad \qquad \qquad Pr(B \, | \, S) = Pr(B \, \cap \, S) \div Pr(S) \\. \\ . \qquad \qquad \qquad \qquad \qquad = \dfrac{1}{5} \div \dfrac{1}{2} \\. \\ . \qquad \qquad \qquad \qquad \qquad = \dfrac{1}{5} \times \dfrac{2}{1} \\. \\ . \qquad \qquad \qquad \qquad \qquad = \dfrac{2}{5} math

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