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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

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You’re not guaranteed to get 5 heads and 5 tails, although that is the most likely option.

Similarly, of you toss the coin a million times, you are not guaranteed to get 500,000 heads, although that is again the most likely combination.

Of course, there are still 499,999 other outcomes you might get.

The 50 percent probability is a *theoretical* probability. Since there are two possible outcomes – heads or tails – then you theoretically have a probability of getting heads or tails half the time.

But theoretical is not reality. If you run the experiment ten times, you’re likely to have a different result – maybe heads 7 times and tails 3 times.

There’s a rule of large numbers which states that the more times you run the experiment, the closer you get to that theoretical probability though. So if you flip a coin 10 times and get 70/30, then flip it 100 times you might get 60/40, and then flip it 1,000 times and you’ll see it come out to 53/47, and then flip it 10,000 times and it’ll be closer to 50.01/49.99

The 50% probability you mention is for a single event. If you’ve flipped a coin once and got Heads, you’re not guaranteed Tails on the next flip, it’s still 50% likely you’ll get Heads.

Similarly, if you’ve flipped 4H and 5T, your 10th flip still has the same 50% likelihood of being Heads or Tails.

So that said, you could find the probability for achieving 5H and 5T. It may be the most likely outcome, but it’s likely less than 50%, because you also have to account for the probability of all the other outcomes: 6/4, 7/3, 8/2, 9/1, 10/0, 4/6, 3/7, 2/8, 1/9, 0/10.

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.

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).

It’s not guaranteed. Each coin toss is independent.

What you end up with is a binomial distribution: [https://mathbitsnotebook.com/Geometry/Probability/BD3x.jpg](https://mathbitsnotebook.com/Geometry/Probability/BD3x.jpg)

It’s likely you will, but not guaranteed. The 50:50 is independent of anything that has come before it, so even if it has landed heads 27 times in a row, the chances of it landing heads again is a still 50:50