# Coin flip question.

Hi I have a question on probability which I am not sure of.

If you have a coin and flip it 20 times then of course the probability of getting H or T is 0.5. And even in 20 flips you should roughly get a 50% distribution, correct?

Now what if to each side of the coin you affix a number, #1 for H, #2 for T.
And before each flip of the coin, you randomly switch the the numbers affixed to each surface.

After flipping the coin another 20 times, both the H-T distribution and the 1-2 distribution should be ~50% right?

Ok now before each coin flip, let's say the numbers 1 and 2 are switched in some non-random way. For example, the rankings of two players in a sport (although I'm not sure if this is considered truly non-random from the perspective of the coin).

Now you flip the coin another 20 times and you still get a ~50% distribution for H-T, but the distribution for 1-2 is very skewed, let's even say 100% '1'. Looking at the H-T distribution it seems normal and random, but looking at the 1-2 distribution one has reason to suspect something is unusual.

Is there a reason to think that someone could have tinkered with the coin flip process to get this skewed distribution or is it not completely abnormal for this very skewed distribution since 1-2 were apparently not switching randomly?

Thanks.

lurflurf
Homework Helper
Yes there is a reason to think that someone could have tinkered with the coin flip process. The number '1' is effectively a guess of the result, one that is 100% accurate. It is very much like if we flipped two coins each time and they always match. It would be very suspicious. In fact the randomness of '1' and '2' is not important, if we change them in any random or deterministic way we should get about 50% if the coin flip is random.

But is that true even if the second coin was not random? i.e. the players rank could have been fixed (since it is non-random) at #1 and #2 for all 20 flips in which case the 1-2 distribution would equal the H-T distribution.

Just replace the tokens #1 and #2 with "this side" and "not this side".
And lurflurf already said that the randomness of #1 and #2 is not important.