Three candidates solve a question. Odds in favour of the correct answer are 5:2, 4: 3 and 3:4 respectively for the three candidates. What is the probability that at least two of them solve the question correctly?

Options

A. 209/343
B. 134/343
C. 149/343
D. 60/343

Answer

A

Explanation

Odd in fav. For student `(A) = (5)/(2) = (P(A))/(P(A))`
Odd in fav. For student `(B) = (4)/(3) = (P(B))/(P(B))`
Odd in fav.for students (C) `= (3)/(4) = (P(C))/(P(C))`.
`rArr P(A) = (2)/(5) P(A) , P(B) = (3)/(4)P(B) ,P(C) = (4)/(3) P(C)`.
Now `P(A) +P(A) = 1 rArr P(A) +(2)/(5) P(A) = 1 rArr P(A) = (5)/(7)`
Also `P(B) +P(B) = 1 rArr P(B) +(3)/(4) P(B) = 1 rArrP(B) = (4)/(7)`
And `P(C) +P(C)= 1 rArr P(C) +(4)/(3)P(c) = 1rArr P(C) = (3)/(7)`
`therefore P(A) = (2)/(5) xx (5)/(7) = (2)/(7)" " P(B)=(3)/(4) xx (4)/(7) = (3)/(7)`
`P(C) = (4)/(3) xx (3)/(7) =(4)/(7)`
Req. Prob. = `P(A) xx P(B) xx P(C) xx +P(A) xx P(B) xx P(C) +P(A) xx P(B) xx P(C)+P(A) xx P(B) xx P(C)`
`= (5)/(7) xx (4)/(7) xx(4)/(7)xx (3)/(7)xx (3)/(7)+(2)/(7) xx (4)/(7) xx (3)/(7)+(5)/(7) xx (4)/(7) xx (3)/(7)`
`= (209)/(343)`