An automated risk-management system uses three independent algorithms—Algorithm X, Algorit… — GRE General Test

An automated risk-management system uses three independent algorithms—Algorithm X, Algorithm Y, and Algorithm Z—to detect fraudulent transactions. The probability that Algorithm X detects a given fraudulent transaction is $\frac{3}{5}$ , the probability that Algorithm Y detects it is $\frac{2}{3}$ , and the probability that Algorithm Z detects it is $\frac{3}{4}$ . If a fraudulent transaction occurs, what is the probability that it will be detected by at least two of these three algorithms?

Cevap: 0.75

Cevap

The probability that the transaction is detected by at least two of the three algorithms is 0.75 (or 3/4).

Because the algorithms operate independently, the event 'at least two algorithms detect the transaction' consists of four mutually exclusive outcomes: exactly X and Y detect (probability 6/60 = 0.10), exactly X and Z detect (probability 9/60 = 0.15), exactly Y and Z detect (probability 12/60 = 0.20), and all three detect (probability 18/60 = 0.30). Summing these four probabilities gives 0.10 + 0.15 + 0.20 + 0.30 = 0.75.

Adım Adım Çözüm

Determine the complementary probabilities of non-detection for each algorithm.

P(X does not detect) = 2/5, P(Y does not detect) = 1/3, and P(Z does not detect) = 1/4.

The probability of an event's complement is 1 minus the probability of the event.

Calculate the probability for each scenario where exactly two algorithms detect the transaction.

P(X and Y only) = 6/60, P(X and Z only) = 9/60, P(Y and Z only) = 12/60.

Since the algorithms operate independently, joint probabilities are calculated by multiplying individual probabilities.

Calculate the probability that all three algorithms detect the transaction.

P(X and Y and Z) = 18/60.

Multiplying the individual detection probabilities of all three independent algorithms.

Sum the probabilities of all qualifying mutually exclusive outcomes.

(6/60) + (9/60) + (12/60) + (18/60) = 45/60 = 0.75.

The events representing different combinations of detections are mutually exclusive, so their probabilities add directly.

Anahtar Kavram

Independent Events and Addition Rule for Mutually Exclusive Outcomes

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