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

1 Answer
MattyMatty
Mar 3, 2018

#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)"#

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