11.6 Conditional Probability 713 The symbol P E E ( ), 2 1 read “the probability of E ,2 given E ,1 ” represents the probability of E2 occurring, assuming that E1 has already occurred (or will occur). Definition: Conditional Probability In general, the probability of event E2 occurring, given that an event E1 has happened (or will happen; the time relationship does not matter), is called conditional probability and is written P E E ( ). 2 1 Example 1 Conditional Probability One card is drawn from a standard 52-card deck. Determine the probability that the card drawn is a a) king. b) king given that the card is a face card. c) diamond. d) diamond given that the card is a red card. Solution a) There are 4 kings in a standard 52-card deck. Therefore, P(king) 4 52 1 13 . = = b) We are asked to determine the probability that the card is a king given that the card is a face card. Since we know the card is a face card, we no longer consider all 52 cards as possible outcomes. Instead we only consider the 12 face cards as possible outcomes. Of these 12 face cards, 4 are kings. Therefore, P(king face card) 4 12 1 3 . = = c) There are 13 diamonds in a standard 52-card deck. Therefore, P(diamond) 13 52 1 4 . = = d) We are asked to determine the probability that the card is a diamond given that the card is a red card. Since we know the card is a red card, we no longer consider all 52 cards as possible outcomes. Instead, we only consider the 26 red cards as possible outcomes. Of these red cards, 13 are diamonds. Therefore, P(diamond red card) 13 26 1 2 . = = 7 Now try Exercise 5 Example 2 Tossing Two Coins Two fair coins are tossed. Determine the probability that a) two tails are tossed. b) two tails are tossed, given that at least one of the tosses is tails. c) two tails are tossed, given that the first toss is tails. Solution a) To determine the probability that two tails are tossed, we can use a tree diagram to determine the sample space of two coin tosses. Then, from the sample space
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