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 the majority are in favour of the book.

Options

A. `35//49`
B. `125//343`
C. `164//343`
D. `209//343`

Explanation

The probability that the first critic favours the book is
`P(E_(1))=(5)/(5+2) =5/7`
The probability that the second critic favours the book is
`P(E_(2))=(4)/(4+3)=4/7`
The probability that the third cirtic favours the books is
`P(E_(3))=(3)/(3+4)=3/7`
Majority will be in favor of the book if at least two critics favour the book. Hence the probability is
`P(E_(1)nnE_(2)nnbarE_(3))+P(E_(1)nnbarE_(2)nnE_(3))`
`" "+P(barE_(1)nnE_(2)nnE_(3))+P(E_(1)nnE_(2)nnE_(3))`
`=P(E_(1)nnE_(2)barE_(3))P(barE_(3))+P(E_(1))P(barE_(2))P(E_(3))`
`" "+P(barE_(1)nnE_(2)nnE_(3))+P(E_(1))P(E_(2))P(E_(3))`
`=5/7xx4/7xx(1-(3)/(7))+5/7xx(1-(4)/(7))xx3/7`
`" "+(1-(5)/(7))xx4/7xx3/7+5/7xx4/7xx3/7=(209)/(343)`