During a quality assurance test of a dual-sensor monitoring device, Sensor A operates inde… — GRE General Test

During a quality assurance test of a dual-sensor monitoring device, Sensor $A$ operates independently of Sensor $B$ . The probability that Sensor $A$ detects a target signal during a test trial is $0.80$ , and the probability that Sensor $B$ detects the target signal during the same trial is $0.75$ . What is the probability that exactly one of the two sensors detects the target signal during a test trial?

Cevap: 0.35

Cevap

0.35

To find the probability that exactly one sensor detects the signal, we must evaluate two disjoint scenarios: Sensor

A

detects while Sensor

B

fails (

0.80 \times 0.25 = 0.20

), or Sensor

A

fails while Sensor

B

detects (

0.20 \times 0.75 = 0.15

). Summing these two probabilities gives

0.20 + 0.15 = 0.35

. Alternatively, subtracting the probability that both sensors detect (

0.80 \times 0.75 = 0.60

) from the probability that at least one detects (

1 - 0.20 \times 0.25 = 0.95

) yields

0.95 - 0.60 = 0.35

Adım Adım Çözüm

Determine the probabilities of individual event non-occurrences

P(A') = 0.20

and

P(B') = 0.25

The probability of an event not occurring is

1

minus the probability that it occurs.

Calculate joint probability for each mutually exclusive event outcome

P(A \text{ and } B') = 0.80 \times 0.25 = 0.20

and

P(A' \text{ and } B) = 0.20 \times 0.75 = 0.15

Since the sensors operate independently,

P(X \text{ and } Y) = P(X) \times P(Y)

Combine the independent outcomes that satisfy the condition

0.20 + 0.15 = 0.35

The scenarios (only

A

detects, or only

B

detects) are mutually exclusive, so their probabilities are added.

Anahtar Kavram

Probability of Independent and Mutually Exclusive Events

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

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