Probability question: hitting the target

Question

The probability of an archer to hit the target is 0.85. The archer keeps shooting until he hits the target, but not more than 5 times. What is the probability that he shot not less than 3 times?

Answer 1

Here's another take on the problem. The archer stops when he/she hits the target. Min. 3 tries. How many tries before hitting the target? Prob. of hit = 0.85; prob. of miss = 0.15.

Misses 1st 2 tries, success on 3rd. Prob = 0.15 x 0.15 x 0.85 = 0.0191

Misses 1st 3 tries, success on 4th. Prob = 0.15 x 0.15 x 0.15 x 0.85 = 0.0029

Misses 1st 4 tries, success on 5th (max no. of attempts). Prob = 0.15 x 0.15 x 0.15 x 0.15 x 0.85 = 0.0004

Sum of probability distribution is 0.0224 or 2.24%

I'd appreciate any comments re: this approach.

Answer 2

P(hitting) = .85 + (1-.85).85 + (1-.85)².85 + ...

or .P(n) = .85(.15)^n-1 (Geometric series)

S_n = .85 x (1-(.15)ⁿ)/(1-.15)

S₅-S₃ or P(5)+P(4)