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Faulty order source

ProbabilityEasy

Desk A sends $60%$ of all orders and desk B sends the other $40%$ . An order from desk A is faulty with probability $5%$ , and an order from desk B is faulty with probability $10%$ , independently across orders. One order is picked at random and it is faulty. What is the probability that it came from desk B? Give a fraction in lowest terms.

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Hint

Compare the joint probabilities of each desk and a faulty order, then normalize.

Solution

The probability of desk A and a faulty order is $0.6 \times 0.05 = 0.03$ .
The probability of desk B and a faulty order is $0.4 \times 0.1 = 0.04$ .
The probability of a faulty order is $0.03 + 0.04 = 0.07$ .
The conditional probability for desk B is $\frac{0.04}{0.07}$ , which is $\frac{4}{7}$ .

Answer: $\frac{4}{7}$

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