Probability

Akansha wrote: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

Hi again Akansha!

Before we start, I want to point out a problem with this particular question. Nowhere does it say that a question has an equal chance of being true or false. Accordingly, it's impossible to answer (and could never appear on the GMAT without that information). We'll go ahead and solve as though that information were included, but be wary of future questions from the same source. Where did you find it?

As always, let's start by analyzing the stem:

each of n questions has 2 possible answers (assumed equally likely, see above); and
need to answer all n correct to win.

The question:

what's the smallest value of n that gives you less than a 1/1000 chance of winning?

Since you're 1/2 likely to be right on each question, the probability of getting n questions correct will be (1/2)^n. Rephrasing the question:

what's the smallest value of n for which (1/2)^n < 1/1000

or

what's the smallest value of n for which 2^n > 1000?

Quick trial and error and an eye on the choices will make the solution super fast:

2^5 = 32, way too small.
2^50 = a REALLY REALLY big number, way too big.

Accordingly, 2^10 must be the correct choice - choose (B)!

Prob of getting 1 question right = 1/2
prob of getting n questions right = (1/2)^n
(1/2)^n <1/1000 => 1/(2^n)<1/1000 => 1000<2^n
2^10 = 1024
and thus n=10

Akansha wrote: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

There are a total of 2^n different ways of registering answers of all n either be true or false type of questions, out of which there’s only one way of registering all n correct, thus the probability of winning is 1/(2^n); and for

1/ (2^n) < 1/1000

2^n > 1000

And the least n for which 2^n > 1000 is [spoiler]10[/spoiler], reason 2^5 = 32 and 32 is the smallest positive integer whose square is more than 1000, that way

32^2 > 1000 or (2^5) ^2 > 1000 or [spoiler]2^10 > 1000

B[/spoiler]

Stuart Kovinsky wrote:

Akansha wrote: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

Hi again Akansha!

Before we start, I want to point out a problem with this particular question. Nowhere does it say that a question has an equal chance of being true or false. Accordingly, it's impossible to answer (and could never appear on the GMAT without that information). We'll go ahead and solve as though that information were included, but be wary of future questions from the same source. Where did you find it?

As always, let's start by analyzing the stem:

each of n questions has 2 possible answers (assumed equally likely, see above); and
need to answer all n correct to win.

The question:

what's the smallest value of n that gives you less than a 1/1000 chance of winning?

Since you're 1/2 likely to be right on each question, the probability of getting n questions correct will be (1/2)^n. Rephrasing the question:

what's the smallest value of n for which (1/2)^n < 1/1000

or

what's the smallest value of n for which 2^n > 1000?

Quick trial and error and an eye on the choices will make the solution super fast:

2^5 = 32, way too small.
2^50 = a REALLY REALLY big number, way too big.

Accordingly, 2^10 must be the correct choice - choose (B)!

Thanks a lot Stuart! I really like the explanation of all your answers. Helps me a lot with the concepts.