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 $\dfrac{14}{15}.$

Step-by-step explanation:  Given that the probabilities that A, B and C can solve a particular problem are $\dfrac{3}{5},~ \dfrac{2}{3},~\dfrac{1}{2}$ 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

$P(E)=\dfrac{3}{5},~~~P(F)=\dfrac{2}{3},~~P(G)=\dfrac{1}{2}.$

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

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

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

$P(\bar{A}\cap \bar{B}\cap \bar{C})=P(\bar{A})\times P(\bar{B})\times P(\bar{C})=\dfrac{2}{5}\times\dfrac{1}{3}\times\dfrac{1}{2}=\dfrac{1}{15}.$

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

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

Thus, the required probability is $\dfrac{14}{15}.$