Exponents/Probability Factors Problem

[jba05d]

N questions can either be true or false. If you answer all n correct you win. What is the least value of n for which the probability is less than 1/1000 for you to win by guessing randomly?
a. 5
b. 10
c. 50
d. 100
e. 1000



I was able to derive the correct answer, which is B. However, I feel that I took a very long approach to derive the answer. I know there is a faster way. They way I basically solved the problem was by setting 2^n = 1000. However, then I literally went through powers until I was close enough to 1000. I know there is a faster way to realize the answer is 10. Could someone please guide me in the right direction. Thx

[SanjeevK]

A question can be either true or false. Let probability of correct answer is p(c) and wrong answer is p(w)
p(c) = 1/2 p(w) = 1/2

Probability of answering n correct answers is: 1/2 * 1/2 * 1/2 ...... n times = 1/(2^n)

Given equation [ 1/(2^n) ] < 1/1000
2^n > 1000

An easy way to solve this is divide 1000 repeatedly till the remainder is zero or 1. This gives value of n as 10 since 1024 is the least number (2^10) > 1000

IMO B