Three critics review a book. Odds in favour of the book are 5:2, 4:3 and 3:4 respectively for three critics. Find the probability that eh majority are in favour of the book.

Explanation

Let A, B, C denote the evets that the book be favoured by the first, second and third critic respectively. Then,
`P(A)=5/7, P(B)=4/7, P(C)=3/7`,
`P(bar(A))=(1-5/7)=2/7, P(bar(B))=(1-4/7)=3/7` and `P(bar(C))=(1-3/7)=4/7`.
Required probability
= P(2 criticec favour the book or 3 critics favour the book)
= P (2 critics favour the book) + P (3 critics favours the book)
= P[{A and B and not C} or {A and C and not B} or {B and C and not A}]+P (A and B and O)
`=P(A nn B nn bar(C))+P(A nn bar(B) nn C)+P(bar(A) nn B nn C)+P(A nn B nn C)`
`={P(A)xxP(B)xxP(bar(C))}+{P(A)xxP(bar(B))xxP(C)}+{P(bar(A))xxP(B)xxP(C)}+{P(A)xxP(B)xxP(C)}`
`=(5/7xx4/7xx4/7)+(5/7xx3/7xx3/7)+(2/7xx4/7xx3/7)+(5/7xx4/7xx3/7)`
`=(80/343+45/343+24/343+60/343)=209/343`.
Hence, the required probability is `209/343`.