why does theoretical probability not align with practice?

Question

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

Answer 1

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!