Question

A boy has 20% chance of hitting at a target. Let p denote the probability of hitting the target for the first time at the nth trial. lf p satisfies the inequality 625p^2 - 175p + 12 <0 then value of n is?

Answer is 3

Answer 1

`n=3`

Explanation:

`p(n) = "Hitting for the 1st time at the n-th trial"`
`=> p(n) = 0.8^(n-1) * 0.2`

`"Boundary of the inequality "625 p^2 - 175 p + 12 = 0"`
`"is the solution of a quadratic equation in "p" :"`
`"disc : "175^2 - 4*12*625 = 625 = 25^2`
`=> p = (175 pm 25)/1250 = 3/25 " or "4/25"`
`"So "p(n)" is negative between those two values."`

`p(n) = 3/25 = 0.8^(n-1) * 0.2`
`=> 3/5 = 0.8^(n-1)`
`=> log(3/5) = (n-1) log(0.8)`
`=> n = 1 + log(3/5)/log(0.8) = 3.289....`

`p(n) = 4/25 = ...`
`=> n = 1 + log (4/5)/log(0.8) = 2`

`"So " 2 < n < 3.289... => n = 3 " (as n is integer)"`