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© Jill Zarestky Math 141 Week in Review Week in Review 7 Key Topics 7.4 Use of Counting Techniques in Probability Let S be a uniform sample space and E be any event in S. Then n(E) number of outcomes in event E P (E) = = number of outcomes in the sample space S n ( S ) • • Use permutations and combinations to find the numbers of outcomes. The rules of probability still apply: o If E and F are mutually exclusive, P(E ∪ F) = P(E) + P(F) o If E and F are any two events, P(E ∪ F) = P(E) + P(F) – P(E ∩ F) o P(EC) = 1 – P(E) 7.5 Conditional Probability and Independent Events Conditional Probability • If E and F are events of an experiment such that P(E) ≠ 0, then the conditional probability P (E ! F) that event F will occur given that event E has occurred is: P ( F | E ) = P (E) • Rearrange to get the product rule: P(E ! F ) = P(E )P(F | E ) • Use tree diagrams to visually represent situations. Independent Events • Two events are independent if the outcome of one does not affect the outcome of the other. • Events E and F are independent if P ( E ! F ) = P ( E ) " P ( F ) . o Use this as your “test.” • If E and F are independent then P ( E | F ) = P ( E ) and P ( F | E ) = P ( F ) . 7.6 Bayes’ Theorem Bayes’ Theorem • • P (E) P (F | E) P (F) Use this to “switch the givens.” P (E | F) = 1