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Difficulty: HardCompound Probability

A secure military facility uses three independent detection systems to monitor its perimeter. The probability that the primary radar detects an incoming drone is 0.80. The probability that the infrared system detects it is 0.90. The probability that the acoustic sensors detect it is 0.75. If a drone approaches the perimeter, what is the probability that exactly two of the three systems detect it? Express your answer as a decimal.

Answer: 0.375

Answer

0.375
The correct answer is found by calculating the compound probability for each of the three specific scenarios where exactly two systems succeed and one fails, and then adding those mutually exclusive probabilities together.

Step-by-Step Solution

1
Identify the mutually exclusive scenarios that result in exactly two systems detecting the drone.
Scenario 1: Radar and Infrared detect, Acoustic fails. Scenario 2: Radar and Acoustic detect, Infrared fails. Scenario 3: Infrared and Acoustic detect, Radar fails.
Because these three systems act independently, we must calculate the compound probability of each specific combination of two successes and one failure.
2
Calculate the probability of Scenario 1 (Radar and Infrared detect, Acoustic fails).
0.80×0.90×(1−0.75)=0.80×0.90×0.25=0.1800.80 \times 0.90 \times (1 - 0.75) = 0.80 \times 0.90 \times 0.25 = 0.180
Multiply the probabilities of the two specific successful detections by the probability of the specific failure.
3
Calculate the probability of Scenario 2 (Radar and Acoustic detect, Infrared fails).
0.80×(1−0.90)×0.75=0.80×0.10×0.75=0.0600.80 \times (1 - 0.90) \times 0.75 = 0.80 \times 0.10 \times 0.75 = 0.060
Multiply the probabilities of the two specific successful detections by the probability of the specific failure.
4
Calculate the probability of Scenario 3 (Infrared and Acoustic detect, Radar fails).
(1−0.80)×0.90×0.75=0.20×0.90×0.75=0.135(1 - 0.80) \times 0.90 \times 0.75 = 0.20 \times 0.90 \times 0.75 = 0.135
Multiply the probabilities of the two specific successful detections by the probability of the specific failure.
5
Sum the probabilities of the three mutually exclusive scenarios.
0.180+0.060+0.135=0.3750.180 + 0.060 + 0.135 = 0.375
To find the total probability of any of these mutually exclusive events occurring, add their individual joint probabilities together.

Key Concept

Calculating the probability of an exact number of successes among independent events by identifying all possible combinations of success and failure.
Estimated Time:2m 30s
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Topics

Compound Probability

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