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P, Q, and R each try to execute a job and create a report on it. The

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New post 06 Jul 2018, 04:00
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e-GMAT Question of the Week #6


P, Q, and R each try to execute a job and create a report on it. The individual probabilities for their completion of the jobs are \(\frac{1}{3}\), \(\frac{2}{3}\), and \(\frac{3}{5}\) and the probability for any of them not to finish the report is \(\frac{2}{5}\). If one can write the report only after finishing the job, then what is the probability that only P and Q will complete their jobs and finish their reports?

    A. \(\frac{4}{625}\)

    B. \(\frac{4}{125}\)

    C. \(\frac{32}{625}\)

    D. \(\frac{4}{25}\)

    E. \(\frac{32}{125}\)


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P, Q, and R each try to execute a job and create a report on it. The  [#permalink]

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New post Updated on: 09 Jul 2018, 20:39
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P, Q, and R each try to execute a job and create a report on it.

Completing the job and finishing the report by P, Q and R are independent of each other.

GIven probability for any of them not finishing the report is \(\frac{2}{5}\)

probability for any of them finishing the report is \(1-\frac{2}{5} = \frac{3}{5}\)

To find out the probability that only P and Q will complete their jobs and finish their reports

We need to find

Required probability is the product of three of the following probabilitites

    P(P will complete his job AND finish the report)
    P(Q will complete his job AND finish the report)
    P(R will not complete his job) OR P(R will complete his job AND not finish the report)

probability that P will complete his job and finish the report = \(\frac{1}{3} * \frac{3}{5} = \frac{1}{5}\)

probability that Q will complete his job and finish the report =\(\frac{2}{3} * \frac{3}{5} = \frac{2}{5}\)

probability that R will not complete his job = \(1 - \frac{3}{5} = \frac{2}{5}\)

probability that R will complete his job and not finish the report = \(\frac{3}{5} * \frac{2}{5} = \frac{6}{25}\)

Substituting the above values, we get \((\frac{1}{5})(\frac{2}{5})(\frac{2}{5} + \frac{6}{25}) = \frac{32}{625}\)

Hence option C
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Originally posted by workout on 06 Jul 2018, 21:40.
Last edited by workout on 09 Jul 2018, 20:39, edited 1 time in total.
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Re: P, Q, and R each try to execute a job and create a report on it. The  [#permalink]

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New post 09 Jul 2018, 06:44
Hey everyone,

We will post the solution very soon. Till then, try it one more time and post your analysis. :-)
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P, Q, and R each try to execute a job and create a report on it. The  [#permalink]

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New post Updated on: 09 Jul 2018, 11:55
Given that:

Probability that \(P\) execute the job = \(\frac{1}{3}\)
Probability that \(Q\) execute the job = \(\frac{2}{3}\)
Probability that \(R\) execute the job = \(\frac{3}{5}\)

Probability that anyone not finishing the report = \(\frac{2}{5}\)
=> Probability that anyone finish the report = 1 - \(\frac{2}{5}\) = \(\frac{3}{5}\)

Given that only \(P\) and \(Q\) will execute the job and finish the report.
So,\(R\) will not execute the job

Therefore Probability of \(R\) not executing the job = 1- \(\frac{3}{5}\) = \(\frac{2}{5}\)

Required probability = (P execute the job and finish report) x ( Q execute the job and finish the report) x (R not executing the job)
= \((\frac{1}{3} * \frac{3}{5}) ( \frac{2}{3} * \frac{3}{5}) (\frac{2}{5})\)
= \(\frac{1}{5} * \frac{2}{5} * \frac{2}{5}\)
= \(\frac{4}{125}\)
Answer = B

Originally posted by siddharthabingi on 09 Jul 2018, 08:09.
Last edited by siddharthabingi on 09 Jul 2018, 11:55, edited 1 time in total.
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P, Q, and R each try to execute a job and create a report on it. The  [#permalink]

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New post 09 Jul 2018, 10:12
I did go with C but the OA says B. So here is why it could be B.

P(E) for P and Q to complete their jobs and reports =
[(P(A)*P(B)*P(C) all completing the Job) * (P(A)*P(B) completing the Report *P(C) not completing the Report)] +
[(P(A)*P(B) completing the Job * P(C) not completing the Job ) * (P(A)*P(B) completing the Report *P(C) not completing the Report)].

We should remember that when C does not complete this job it also implies that he cannot complete his report.

Substituting the values we get ->[\((1/3*2/3*3/5)* (3/5*3/5*2/5) = 12/625\)] + [\((1/3*2/3*2/5)* (3/5*3/5*2/5) = 8/625\)] = \(20/625\) = \(4/125\)

Do let me know if my understanding is wrong.

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Re: P, Q, and R each try to execute a job and create a report on it. The  [#permalink]

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New post 09 Jul 2018, 11:42
The reason why I calculated B and not C is the last part of the question: then what is the probability that only P and Q will complete their jobs and finish their report For me completing the job and finishing their report implies that R was not able to complete the job. Therefore, it is not necessary to take into account, that R could have completed the job but not the report.
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New post 12 Jul 2018, 01:08

Solution



Given:
    • P, Q, and R each try to execute a job and create a report on it
    • The individual probabilities for their completion of the jobs are \(\frac{1}{3}\), \(\frac{2}{3}\), and \(\frac{3}{5}\)
    • The probability for any of them not to finish the report is \(\frac{2}{5}\)
      o Hence, the probability of finishing the report = \((1 – \frac{2}{5}) = \frac{3}{5}\)
    • One can write the report only after finishing the job

To find:
    • The probability that only P and Q will complete their jobs and finish their reports

Approach and Working:

it is given that, as per the favourable event, only P and Q will complete their jobs and finish their reports

    • The probability that P will execute the job and finish the report = \(\frac{1}{3} * \frac{3}{5} = \frac{1}{5}\)
    • The probability that Q will execute the job and finish the report = \(\frac{2}{3} * \frac{3}{5} = \frac{2}{5}\)

Now, as only P and Q will complete their jobs and finish their reports, it also means, there are two possibilities exist for R

    • Either R will not finish the job, and definitely not finish the report (as one cannot finish the report without executing the job)
      o Probability of this event = \((1 – \frac{3}{5}) * \frac{2}{5} = \frac{2}{5} * \frac{2}{5} = \frac{4}{25}\)
    • Or else, R will execute the job but will not finish the report
      o Probability of this event = \(\frac{3}{5} * \frac{2}{5} = \frac{6}{25}\)

So, we can say,
    • Probability (only P and Q will complete their jobs and finish their reports) =
    P (P completes job & finish report) AND P (Q completes job & finish report) AND [P (R does not complete the job & does not complete the report) OR P (R does complete the job & does not complete the report)
    = \(\frac{1}{5} * \frac{2}{5} * [\frac{4}{25} + \frac{6}{25}]\)
    = \(\frac{1}{5} * \frac{2}{5} * \frac{2}{5}\)
    = \(\frac{4}{125}\)

Hence, the correct answer is option B.

Answer: B
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