Question

Alan, Betty and Carol are applying for three different jobs in a company. The probability that Alan gets his job is 0.5; the probability that Betty gets her job is 0.75; the probability that Carol gets her job is 0.3. Can anyone solve this?

What is the probability that Jobs will be obtained by

a) None of them? (e) Two of them?
b) Only one of them? (f) At least two of them
c) At least one of them? (g) At most two of them
d) At most one of them? (h) All three of them?

Answer 1

see below (long answer ahead)

Explanation:

the probability that alan gets his job is `0.5`; the probability that he doesn't is `1-0.5`, which is `0.5`.

the probability that betty gets her job is `0.75`; the probability that she doesn't is `1-0.75`, which is `0.25`.

the probability that carol gets her job is `0.3`; the probability that she doesn't is `1-0.3`, which is `0.7`.

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if none of them get their job, this means that alan doesn't get his, betty doesn't get hers, and carol doesn't get hers.

the probability of all of these happening at the same time are calculated by multiplying the probabilities together. here, `0.5`, `0.25` and `0.7` are multiplied to get `0.0875`.

the probability that none of them get their jobs is `0.0875`.

in the same way, the probability that all of them do get their jobs is `0.5 * 0.75 * 0.3`, which is `0.1125`.

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for only one of them to get the job, either alan, betty or carol could get their job while the other two wouldn't.

the probability of only alan getting the job would be `0.5 * 0.25 * 0.7`, which is `0.0875`
the probability of only betty getting the job would be `0.5 * 0.75 * 0.7`, which is `0.2625`
the probability of only carol getting the job would be `0.5 * 0.25 * 0.3`, which is `0.0375`
the probability of one of these happening would be `0.0875 + 0.2625 + 0.0375`, which is `0.3875`

for two of them to get the job, either alan and carol, betty and carol or betty and alan could get their jobs while the other person wouldn't.

the probability of only alan and carol getting jobs would be `0.5 * 0.25 * 0.3`, which is `0.0375`
the probability of only betty and carol getting jobs would be `0.5 * 0.75 * 0.3`, which is `0.1125`
the probability of only alan and betty getting jobs would be `0.5 * 0.75 * 0.7`, which is `0.2625`
the probability of one of these happening would be `0.0375 + 0.1125 + 0.2625`, which is `0.4125`

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for at least one of them to get a job, the only thing that cannot happen is that none of them get a job.

the probability that none of them do is `0.0875`.
the probability that this does not happen is `1 - 0.0875`, which is `0.9125`.

for at most two of them to get a job, the only thing that cannot happen is that all of them get a job.

the probability that all of them do is `0.1125`.
the probability that this does not happen is `1 - 0.1125`, which is `0.8875`.

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for at most one to get a job, either none of them get a job or only one person does.

the probability of no one getting a job is `0.0875`
the probability of one person getting a job is `0.3875`

`0.0875 + 0.3875 = 0.475`

for at least two to get a job, either all of them get a job or only two people do.

the probability of all of them getting a job is `0.1125`
the probability of two people getting a job is `0.4125`

`0.1125 + 0.4125 = 0.525`