The probability that a biased coin falls heads is 1/4. What is the lea

1- Probability(No Head) > 0.999

Probability(No Head) < 0.001

Probability (Head) = 1/4

Probability (No Head) = 3/4

(3/4)^N < 0.001

N = 25

Hope this helps

The equation is easy to get, but the question here is how you solve it without calculator.

How do we get the value of n once we have the equation ready, without a calculator Bunuel

How did you calculate this.
TIA

How do we simplify (3/4)^N < 0.001 part?

There's no chance at all you'd ever need to work with something like this on the GMAT, but we want the smallest value of n for which (3/4)^n < 1/1000. If you notice:

3^5 = 243
4^5 = 1024

you can see that 3^5/4^5 < 1/4. So raising both sides of this inequality to the fifth power,

(3^5/4^5)^5 < (1/4)^5
3^25/4^25 < 1/1024 < 1/1000

so that proves the answer here is at most 25, which gets us down to two answer choices. But (3/4)^24 is just negligibly greater than 1/1000 (here I'm using a calculator), so no similar estimation technique would ever let you rule out answer A, and I would be very surprised if there was any practical way using pen and paper to get the right answer here. You almost certainly need to do the calculation exactly, which isn't practical by hand. Not sure of the source, but this is not a GMAT question.
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Approximately Log 2 = .30 log = 4 = .6 , log .475

-3 = N (.475 - .60)
-3 = N (-.125)

so N more close to 24 hence difficult