Respuesta :
Answer: The required probability is [tex]\dfrac{14}{15}.[/tex]
Step-by-step explanation: Given that the probabilities that A, B and C can solve a particular problem are [tex]\dfrac{3}{5},~ \dfrac{2}{3},~\dfrac{1}{2}[/tex] 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
[tex]P(E)=\dfrac{3}{5},~~~P(F)=\dfrac{2}{3},~~P(G)=\dfrac{1}{2}.[/tex]
So, the probabilities that the problem is not solved by A, not solved by B and not solved by C are given by
[tex]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}.[/tex]
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
[tex]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}.[/tex]
Therefore, the probability that at least one of the group solves the problem is
[tex]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}.[/tex]
Thus, the required probability is [tex]\dfrac{14}{15}.[/tex]
