Problem Solving

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

OA is B

Probability of getting a correct answer on each question is 1/2. (1 correct answer out of 2 possible outcomes.)

If the test had 2 questions, the probability of winning would have been 1/2 * 1/2 = 1/4.

In this way, we got to find out at what stage the probability goes over 1/1000

(1/2) ^ 9 = 1/512
(1/2) ^ 10 = 1/1024
Which means if there are 10 questions, the probability is less than 1/1000 (1/1000 > 1/1024)

Hence B

Since there are only two choices and you're guessing randomly the chance of getting a question right is
1/2
So we want to solve (1/2)^n<1/1000.

2^9=512 and 2^10=1024, so n must equal 10.

I have found that it's useful to memorize exponents of 2 all the way through 10 since they show up so often. 2^10 is easy to remember because it's the first exponent of two to break 1000. If you have that memorized, you should spot this answer right away.

Just want to second SoCan's recommendation to memorize 2^n up through 2^10. Indeed they DO show up a lot -- in any problems with equally likely binary setups (like coin-flipping problems), as well as in many problems where populations (of bacteria, say) are doubling every fixed amount of time.

AND, since
2^4 = (2^2)^2, and
2^6 = (2^3)^2, and
2^8 = (2^4)^2, and so on...
those three come free of charge if you've already got 4^2, 8^2, and 16^2 memorized , and also provide a handy reminder of our exponent rule that x^(ab) = (x^a)^b and vice versa.

For one question the probability when the ans is true : 1/2
when the ans is not true : 1/2


The entire probability = 1/(2)^n where n is the number of questions

The min value of n for which the entire probability is less than 1/1000 is 10. i.e Probability =1/1024

So n= 10.

I basically used exponents on each of the options and worked my way down.

A: 1/2^5 = 1/64 (too big)
B: 10..... 1/2^8 = 1/512. So 1/2^10 will be much less than 1/1000.

We have found that the least value for n is 10.

Answer: B