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A particular task is given to three persons, Dave, Giles and Martha whose probabilities of completing it are $\dfrac{3}{4}$ , $\dfrac{2}{8}$ and $\dfrac{3}{8}$ respectively, independent of each other. What is the probability that the task will be completed?

$\textbf{My approach:}$

$\textbf{Let } P(D) = \dfrac{3}{4}$

$\textbf{Let } P(G) = \dfrac{2}{8}$

$\textbf{Let } P(M) = \dfrac{3}{8}$

$\textbf{Probability of task will be completed is :} P(D \cap G \cap M) = \dfrac{3}{4}.\dfrac{2}{8}.\dfrac{3}{8} = \dfrac{9}{128}$

Is my solution and answer is correct?

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3 Answers 3

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What you calculated was the probability that all 3 will complete the task. You should calculate the probability that no one will then subtract from 1.

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  • $\begingroup$ You should calculate the probability that no one will - I need help in understanding this, the question does ask what is the probability the task will be completed, so why subtract it from 1? $\endgroup$
    – Abbas
    Jun 29, 2022 at 15:00
  • $\begingroup$ There are two possible outcomes: someone completes or no one does. You can find the someone-does probability by adding the probabilities of the case where all complete, the three cases where 2 complete, and the three cases where 1 completes. It's much easier to calculate the 1 case where no one does. Then you subtract that probability from 1 (the sum of all 8 cases). You find the outcome where someone completes by subtracting the case where no one does. $\endgroup$
    – stretch
    Jun 30, 2022 at 17:59
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Continuing from @stretch’s answer,

$P(D$ doesn’t complete$)=1-\frac 34=\frac 14$
$P(G$ doesn’t complete$)=1-\frac 28=\frac 68=\frac 34$
$P(M$ doesn’t complete$)=1-\frac 38=\frac 58$.

$\displaystyle P($none of them completes) $\displaystyle =P(D^c\cap G^c\cap M^c)=P(D^c)P(G^c)P(M^c)=\frac 14\cdot\frac 34\cdot\frac 58=\frac{15}{128}.$

By de Morgan’s Law, $D^c\cap G^c\cap M^c=(D\cup G\cup M)^c$
So $P($at least one of them completes)$\displaystyle=P(D\cup G\cup M)=1-P((D\cup G\cup M)^c)=1-P(D^c\cap G^c\cap M^c)=1-\frac{15}{128}=\frac{113}{128}.$

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  • $\begingroup$ Still I wonder why do we need to calculate the probability of no one completing! $\endgroup$
    – Abbas
    Jun 29, 2022 at 16:09
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    $\begingroup$ Because at least one person completing, and no one completing, are exactly complementary events. $\endgroup$
    – insipidintegrator
    Jun 29, 2022 at 16:10
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Your calculation looks correct. The intersection of three independent probabilities would ultimately be the multiplication of the three probabilities.

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    $\begingroup$ This is wrong, he is instead calculating the probability that all three people will complete the task. $\endgroup$
    – jtb
    Jun 29, 2022 at 14:54

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