A computer repair shop has estimated the probability that a computer sent to the shop has a bad modem is 1/8, the probability that the computer has a bad CPU is 1/5, and the probability that it has a bad drive is 1/3. If we assume that modems, CPUs, and drives are independent, find the probability of the following:

a) A modem, CPU, and a drive in a computer sent to the shop are bad
b) Only a modem and a CPU in a computer sent to the shop are bad
c) None of the three parts (modem, CPU, or drive) is bad

1-4

1-5
1/6

To find the probabilities of the given scenarios, we'll use the concept of independent events and the multiplication rule.

a) To find the probability that a modem, CPU, and drive in a computer are bad, we multiply the probabilities of each individual part being bad. Since the parts are assumed to be independent, the probability of all three being bad is:

P(modem, CPU, drive being bad) = P(modem is bad) × P(CPU is bad) × P(drive is bad)

Given:
P(modem is bad) = 1/8
P(CPU is bad) = 1/5
P(drive is bad) = 1/3

P(modem, CPU, drive being bad) = (1/8) × (1/5) × (1/3) = 1/120

Therefore, the probability of a modem, CPU, and drive in a computer being bad is 1/120.

b) To find the probability that only the modem and CPU in a computer are bad, we need to consider that the drive is not bad. Again, assuming independence, we have:

P(modem and CPU are bad, drive is not bad) = P(modem is bad) × P(CPU is bad) × P(drive is not bad)

P(drive is not bad) = 1 - P(drive is bad) = 1 - (1/3) = 2/3

P(modem and CPU are bad, drive is not bad) = (1/8) × (1/5) × (2/3) = 1/60

Therefore, the probability of only the modem and CPU being bad, with the drive being not bad, is 1/60.

c) To find the probability that none of the three parts (modem, CPU, or drive) is bad, we need to find the complementary probability of any part being bad. The probability that a part is not bad is 1 minus the probability of the part being bad.

P(none of the parts is bad) = 1 - P(modem is bad) × P(CPU is bad) × P(drive is bad)

P(none of the parts is bad) = 1 - (1/8) × (1/5) × (1/3) = 119/120

Therefore, the probability that none of the three parts (modem, CPU, or drive) is bad is 119/120.

To find the probabilities, we can use the multiplication rule for independent events:

a) The probability that all three parts (modem, CPU, and drive) are bad can be calculated as the product of the individual probabilities:

P(modem and CPU and drive) = P(modem) * P(CPU) * P(drive)

P(modem and CPU and drive) = (1/8) * (1/5) * (1/3)

b) The probability that only the modem and CPU are bad can be calculated as the product of the individual probabilities:

P(modem and CPU and not drive) = P(modem) * P(CPU) * P(not drive)

Since the events are independent, the probability of not having a bad drive is equal to 1 minus the probability of having a bad drive:

P(not drive) = 1 - P(drive)

P(modem and CPU and not drive) = (1/8) * (1/5) * (1 - 1/3)

c) The probability that none of the three parts (modem, CPU, or drive) is bad can be calculated as the complement of having any of them bad:

P(none bad) = 1 - P(at least one bad)

P(none bad) = 1 - [P(modem) + P(CPU) + P(drive) - P(modem and CPU) - P(modem and drive) - P(CPU and drive) + P(modem and CPU and drive)]

P(none bad) = 1 - [(1/8) + (1/5) + (1/3) - (1/8) * (1/5) - (1/8) * (1/3) - (1/5) * (1/3) + (1/8) * (1/5) * (1/3)]

I hope this helps! Let me know if you have any further questions.