Question

the probabilities that a b and c can solve a particular problem are 3/5 2/3 and 1/2 respectively if they all try determine the probability that at least one of the group solves the problem

Answer 1

Answer: The required probability is
`(14)/(15).`

Step-by-step explanation: Given that the probabilities that A, B and C can solve a particular problem are
`(3)/(5),~ (2)/(3),~(1)/(2)` respectively.

We are to determine the probability that at least one of the group solves the problem , if they all try.

Let E, F and G represents the probabilities that the problem is solved by A, B and C respectively.

Then, according to the given information, we have

`P(E)=(3)/(5),~~~P(F)=(2)/(3),~~P(G)=(1)/(2).`

So, the probabilities that the problem is not solved by A, not solved by B and not solved by C are given by

`P\bar{(A)}=1-P(A)=1-(3)/(5)=(2)/(5),\\\\\\P\bar{(B)}=1-P(B)=1-(2)/(3)=(1)/(3),\\\\\\P\bar{(C)}=1-P(C)=1-(1)/(2)=(1)/(2).`

Since A, B and C try to solve the problem independently, so the probability that the problem is not solved by all of them is

`P(\bar{A}\cap \bar{B}\cap \bar{C})=P(\bar{A})* P(\bar{B})* P(\bar{C})=(2)/(5)*(1)/(3)*(1)/(2)=(1)/(15).`

Therefore, the probability that at least one of the group solves the problem is

`P(A\cup B\cup C)\\\\=1-P(\bar{A\cup B\cup C})\\\\=1-P(\bar{A}\cap \bar{B}\cap \bar{C})\\\\=1-(1)/(15)\\\\=(14)/(15).`

Thus, the required probability is
`(14)/(15).`