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

It’s not that unlikely that you’ll get 5 heads and 5 tails. It’s a 24.6% chance which is the most of any of the possibilities.

Here are the odds:

* 5H, 5T – 24.6%
* 4H, 6T – 20.5%
* 6H, 4T – 20.5%
* 3H, 7T – 11.7%
* 7H, 3T – 11.7%
* 2H, 8T – 4.4%
* 8H, 2T – 4.4%
* 1H, 9T – 1.0%
* 9H, 1T – 1.0%
* 10T – 0.1%
* 10H – 0.1%

So, although 24.6% (about 1 in 4) does not seem that high, it is still the most common *exact* outcome and all the other outcomes are more or less likely based on how far they are from that. There is about a 2 in 3 shot that the number of heads will be between 4 and 6 and about a 9 in 10 chance that the number of heads will be between 3 and 7.

Fun fact is that that distribution gets tighter the more times you roll. But ten is not all that many.