A company has developed an automated security system to detect intruders at a business park. In testing, the system is observed under three mutually exclusive weather conditions (clear, cloudy, and rainy). When an intruder is present, the likelihood of detection depends of the weather condition. If the weather is clear, the intruder is detected 94 percent of the time; If the weather is cloudy, the intruder is detected 80 percent of the time; and if the weather is rainy, the intruder is detected 64 percent of the time.
The three weather conditions occur according to the following percentages: Clear: 50 percent; Cloudy: 40 percent; and Rainy: 10 percent.
Enter answers as decimals below. Round to three decimal places.
(a) If an occasion in which an intruder was present is selected at random, what is the probability that the intruder will be detected? .
(b) If an intruder was detected, what is the probability that the weather was clear? .
(c) If an intruder was detected, what is the probability that the weather was cloudy? .
(d) If an intruder was detected, what is the probability that the weather was rainy? .
Ans:
P(detected/clear)=0.94
P(detected/cloudy)=0.80
P(detected/rainy)=0.64
a)
P(detected)=0.94*0.5+0.80*0.4+0.64*0.1=0.854
b)
P(clear/detected)=P(detected/clear)*P(clear)/P(detected)=0.94*0.5/0.854=0.550
b)
P(cloudy/detected)=P(detected/cloudy)*P(cloudy)/P(detected)=0.8*0.4/0.854=0.375
c)
P(rainy/detected)=P(detected/rainy)*P(rainy)/P(detected)=0.64*0.1/0.854=0.075
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