Probability of a task getting completed if assigned to three persons.

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?

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.

• 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? Jun 29, 2022 at 15:00
• 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. Jun 30, 2022 at 17:59

$$P(D$$ 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)$$\displaystyle=P(D\cup G\cup M)=1-P((D\cup G\cup M)^c)=1-P(D^c\cap G^c\cap M^c)=1-\frac{15}{128}=\frac{113}{128}.$$

• Still I wonder why do we need to calculate the probability of no one completing! Jun 29, 2022 at 16:09
• Because at least one person completing, and no one completing, are exactly complementary events. Jun 29, 2022 at 16:10

Your calculation looks correct. The intersection of three independent probabilities would ultimately be the multiplication of the three probabilities.

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