714 CHAPTER 11 Probability A number of formulas can be used to determine conditional probabilities. We will use the following formula. we can determine the probability that both tosses are tails. The tree diagram and the sample space are shown in Fig. 11.17. There are four possible equally likely outcomes: HH, HT, TH, and TT. Only one of the outcomes has two tails, TT. Thus, P(two tails ) 1 4 = b) We are given that at least one of the tosses is a tail. Therefore, our sample space now becomes HT, TH, TT. There are three possibilities, of which only one has two tails, TT. Thus, P(two tails at least one tail) 1 3 = c) We are given that the first toss is tails. Therefore, the sample space now becomes TH, TT. There are two possibilities, of which only one has two tails, TT. Thus, P(two tails first is tails) 1 2 = 7 Now try Exercise 37 Sample Space 2nd Coin 1st Coin HH TH HT TT H H T T T H Figure 11.17 Conditional Probability For any two events, E1 and E ,2 the conditional probability, P E E ( ), 2 1 is determined as follows. = P E E n E E n E ( ) ( and ) ( ) 2 1 1 2 1 In the formula, n E E ( and ) 1 2 represents the number of sample points common to both event 1 and event 2, and n E( )1 is the number of sample points in event E ,1 the given event. Since the intersection of E1 and E ,2 symbolized E E , 1 2 ∩ represents the sample points common to both E1 and E ,2 the formula can also be expressed as = ∩ P E E n E E n E ( ) ( ) ( ) 2 1 1 2 1 Fig. 11.18 is helpful in explaining conditional probability. U E1 E2 E1 E2 a b c d e f g h i Figure 11.18 MATHEMATICS TODAY Chance of Showers At one time or another, you probably have been caught in a downpour on a day the weather forecaster had predicted sunny skies. Short-term (24-hour) weather forecasts are correct about 90% of the time, and long-term (7-day) forecasts are correct about 80% of the time. This level of accuracy is achieved through the use of conditional probability. Computer models are used to analyze data taken on the ground and in the air and make predictions of atmospheric pressures at some future time, say 10 minutes ahead. Based on these predicted conditions, another forecast is then computed. This process is repeated until a weather map has been generated for the next 12, 24, 36, and 48 hours. Since each new prediction relies on the previous prediction being correct, the margin of error increases as the forecast extends further into the future. Why This Is Important Weather forecasting is one of many real-life applications of conditional probability. Anna Om/Shutterstock
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