eli5 If an unbiased coin has a 50% chance of getting heads and 50% of getting tails. If you roll 10 times why is it unlikely for you to get 5 heads and 5 tails. From a probability standpoint, shouldn’t you guarantee that?

Question

eli5 If an unbiased coin has a 50% chance of getting heads and 50% of getting tails. If you roll 10 times why is it unlikely for you to get 5 heads and 5 tails. From a probability standpoint, shouldn’t you guarantee that?

In: Mathematics

Answer 1

Probability doesn’t guarantee anything (see the 2016 US election). That’s what “probably” means. On AVERAGE you will end up with 5 heads and 5 tails if you do the test enough. Now every coin flip is independent, so if you look at the total possibilities of what can happen, you end up with 1024 possibilities (2 possibilities for the first, 2 for the second etc.)

Now 210 contain 4 heads and 6 tails which is less likely than 5/5, but a different 210 contain 6 heads and 4 tails. Therefore with 420 possible outcomes, you’re more likely to end up with 4 and 6 about which you want than 5 and 5. So if you’re asking about getting a specific set of results, vs any other result, of which there are many more possibilities.

As an example (and sorry if markup sucks or you’re on mobile), I’ll do this for 4 flips, where we have 16 total possibilities:

HHHH (1)

HHHT HHTH HTHH THHH (4)

HHTT HTTH HTHT THTH THHT TTHH (6)

HTTT THTT TTHT TTTH (4)

TTTT (1)

So even though the most common outcome is two heads and two tails with (6), there are overall more possibilities that are something other than 2 heads and 2 tails with (10).