Flip fair coin n times to get $100 if 60% or over lands heads. What should n be?

Question

If we flip a fair coin n times and 60% or more of those flips lands heads, $100 is given. Do I want the coin to be flipped 100, 1000, or 1M times?

Answer 1

If you allowed to flip once, probability of winning $100 is 50%

For 2 flips, only HH win the pot. $\quad P = {1 \over 4} = 25\%$
For 4 flips, ways $=\binom{4}{4} + \binom{4}{3} = 1+4 = 5,\quad P = {5 \over 16}=31.25\%$
For 6 flips, ways $=\binom{6}{6} + \binom{6}{5} + \binom{6}{4} = 1+6+15 = 22, ,\quad P={22 \over 64} =34.375\%$

It seems chance of winning goes up, but no.
The denominator grows much faster than the numerator.

For 100 flips, $P = \large{\sum_{k=60}^{100}\binom{100}{k} \over 2^{100}} \normalsize ≈ 2.84\%$

For 1000 flips, doing the same math, $P ≈ 0.0000000136\%$

I'll pick the 100 flips.

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Trivia: Besides 1 flip, 3 flips and 5 flips also gives you 50% chance of winning !

Answer 2

The Law of Large Numbers states that, as a sample size grows, its mean will get closer and closer to that of the whole population. Basically, the more you flip the coin, the more the experimental probability approaches the theoretical probability. Therefore, you should pick the smallest number of flips as the larger numbers will be closer to a $50/50$ chance.