A fair coin is tossed n-times such that the probability of getting at least one head is at least 0.9. Then the minimum value of n is ______________.

Solution

Maths sir · 2 steps

At least one head is everything except all tails, so work with the complement (1/2)ⁿ.

1. Use the complement. The only way to get no head is to get a tail on every toss. That has probability (1/2)ⁿ, so P(at least one head) = 1 - 1/2ⁿ.

   1 - 1/2ⁿ ≥ 0.9 ⇒ 1/2ⁿ ≤ 0.1 ⇒ 2ⁿ ≥ 10

   n	2ⁿ	1 - 1/2ⁿ	At least 0.9?
   2	4	0.75	No
   3	8	0.875	No
   4	16	0.9375	Yes

2. So. 2³ = 8 is less than 10, but 2⁴ = 16 is at least 10. The minimum value of n is 4.

The common slip: 3. Rounding 0.875 up to 0.9 gives 3, but 0.875 is below 0.9, so three tosses are not enough.

The idea to keep. Whenever a question says "at least one", find the probability of "none" and subtract it from 1. Then test the smallest whole numbers one by one.

Answer from the NTA final answer key. Solution by PrepWiser.