For example, when I flip a coin, I have a 1/2 chance of getting head, and the same chance at getting tails. With that theory, if I toss a coin 50 times, I should get 25 heads, and 25 tails.
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However, I did 3 rounds of 50 coin flips, with results of:
1. 28 heads, 22 tails
2. 27 heads, 23 tails
3. 27 heads, 23 tails.
I assumed that perhaps the coins weren’t “true”. Maybe the sides are waited different, maybe I tossed with different height/force each time. So I went virtual, and switched to having a computer roll a dice.
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I should have a 1/6 chance at rolling a number between 1-6. So 60 rolls, should have each number come up 10 times. But in practice my results were:
1. 14
2. 9
3. 8
4. 13
5. 6
6. 10
So how come practice/reality, doesn’t align with theory, even when we take human error out of the equation?
In: 34
It’s not theoretical probably it’s the actual expected value.
Let’s go over flipping coins.
The expected value of a fair coin toss with heads = 1 and tails = 0 is 0.5. We calculate that by weighting each outcome 1 and 0 by 0.5(the probability of each outcome) the expected value of a biased coin that flips heads 75% of the time is .75.
Okay we have this average value for the number of heads per flip. How can we show that our assumption that the coin is fair is true?
Well, you can get an estimate of the expected value by drawing from a random distribution and taking the mean (value/number of draws)[aka average] This is a good startagy because its been proven this will converge to the expected value as the number of draws goes to infinity. This is so useful we even know the expected value of the error of this estimate of the expected value! We call it the variance!
So, why won’t my coins estimated expected value equal the theoretical one. Well, there is error associated with your measurement. Maybe the coin is biased or maybe your measurement is within your measurement error of a fair coin. In which case all you can do is lower the error bars on your measurement untill you are statisfied by flipping more coins!
It’s simple. You didn’t flip it enough.
The more flips you do the closer and closer it will get to 50/50. After an infinite number of flips it will even out.
Your theory is based on 50 flips, with a 50/50 outcome, should be 25 to each side.
However your practice isn’t the same test. You’re doing 50 individual 1/2 tests.
If it’s easier imagine a bag of 50 marbles, 25 are red, 25 are blue.
In your theory, your pulling out the marbles one at a time and marking down the colour, but then you leave the marble. Out of the bag, and at the end you’ll have 50 marbles, 25 are red 25 are blue.
In your practice, you’re pulling out a marble noting if it’s red or blue, and then putting it back in the bag before pulling out a marble and marking if it’s red or blue.
It’s not the same test.
Ask yourself a slightly different question. What’s the probability of throwing exactly 25/25?
Turns out it’s only [about 1 in 9](https://www.wolframalpha.com/input?i=50+coin+tosses).
It’s easier to see with smaller numbers. With two coins, there’s four ways they can fall: HH, HT, TH, TT. Half the time, you’ll see an even 1/1 split. With four coins, you have sixteen results, of which six are even 2/2 splits, for an overall 3/8 (37.5%) probability:
HHHH, HHHT, HHTH, HHTT,
HTHH, HTHT, HTTH, HTTT,
THHH, THHT, THTH, THTT,
TTHH, TTHT, TTTH, TTTT
In general, the more coins you flip, the less likely you’ll get an exact 50/50 split.
The theory aligns perfectly with practice, you are just trying to use the wrong theory on a practical situation it doesn’t apply to.
The theory says: every time you flip a coin*, it will either land on head or on tail.
Special attention to “every time” – this theory applies to each individual flip, but says nothing about a series of flips.
And indeed in practice, every time the coin will land either on head or on tail.
The theory providing an answer to your question here would go something like: out of 50 flips, 50!/(x!*(50-x)!)% will be heads. Here x is the number of heads, and the ‘!’ means factorial operation.
In other words, probabilities don’t predict the future.
*a coin that can’t land on it’s edge
Because of statistics. The coin flip it’s a 50/50 chance each flip. Not 50/50 over all flips. Theoretical is a guestimate, a theory. Sometimes we’re exactly right with theories. Most times not.
> If I toss a coin 50 times, I should get 25 heads, and 25 tails
It is not a guaranteed and the whole statistics field is more or less about calcs around the thing that it is not guaranteed
For a real eli5 explanation, you aren’t doing it enough times . Probability of outcomes are based on if you repeat the experiment infinite times. If I toss a coin once and it comes out heads, I can’t say the odds aren’t 50-50. If I toss it again and it’s heads I can’t say it’s not 50-50. If I did it 1,000,000 times, I’d start seeing a pattern emerge
Because of randomness. Theoretical probability is only guaranteed to happen after infinite attempts.
If I were to flip a coin 2 times, and I was guaranteed to get one heads and one tails, then after the first flip, I get heads, the. I would know with absolute certainty that the next one would be tails, and that doesn’t make sense because the next flip should be a 50/50 between heads and tails.
Theoretical probability DOES align with practice. The problem is you’re misattributing something to theoretical probability that it doesn’t say itself.
It should be obvious if you flip a coin twice, you are not guaranteed to get 1 heads and 1 tails, as you can also get 2 heads or 2 tails. As obvious as this may sound, you cannot guarantee what a coin is going to land on. Since you cannot guarantee the outcome of the coin flip, you can’t guarantee the outcome of multiple coin flips. Thus, you can’t guarantee 25 heads and 25 tails.
Probability can only tell you how likely something is to happen. It cannot tell you what will happen (barring a 100% or 0% probability).
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