I have a truly burning question.
If you pick a number 1-10 100 times completely randomly and each number has a 10% chance of being picked each time (so picking a 7 once doesn’t decrease the likelihood of picking it on the next try) why won’t you end up with 10 of each number?
In: 1
You would end up with approximately 10 of each number. But the chances of getting *exactly* 10 of each number are still very small.
Because probability doesn’t predict what will happen for exactly n trials. It predicts what will happen over *many* trials, where *many* means, really, the limit as n goes to infinity. The more and more times you repeat the random trial of picking a number from 1 to 10, the closer and closer you’ll get to having 10% of the total from each number. But you only get *exactly* 10% of each in the limit, with infinitely many trials (which is of course impossible to do in real life)
It may be easier to see what’s happening by taking a simpler case. Take a fair coin with a 50% chance of landing (H)eads and 50% chance of landing (T)ails and flip the coin 4 times.
After 4 flips, there are 16 possible outcomes:
HHHH / HHHT / HHTH / HHTT / HTHH / HTHT / HTTH / HTTT / THHH / THHT / THTH / THTT / TTHH / TTHT / TTTH / TTTT
Given the same reasoning as in your question, we would expect that for a coin flipped with equal chance of outcomes, we would have 2 Heads and 2 Tails after each flip. In reality, only 6 of the possible 16 cases have this configuration.
This same principle can be applied to your original question. I won’t do the math on it because the total number of possible outcomes is (literally) 10^100 but it works the same way.
Hope this helps.
When you pick a number with a 10% chance, the chances of picking the second number, for each number, is not 10% anymore, it’s 10% x 10% which makes it a 1% chance. To ELY5, you pick 1 and then you have 10 choices for the second number, you pick 2 and then 10 choices for the second number… We have 10 choices for the first number and each choice has 10 choices for the second number. 10×10=100. So the chances of picking a 7 and then any number is 1%. Chances of picking a 7 and then a 7? 1%. By picking more numbers, you make more probabilities. Each step you add to pick a number gives you 10 new possibilities per each possibility you had before. So, if you pick a 7, the chances of picking a number between 1-10 is also 1%, however, the chances of picking 7 and 7 again is 1%, but the chances of picking 7 and then any number but 7, is 9%, which is much higher than 1%. (PS: If you want to pick the number 7 a hundred times, the probability of it would be 1 in 10^100)
> If you pick a number 1-10 100 times completely randomly and each number has a 10% chance of being picked each time (so picking a 7 once doesn’t decrease the likelihood of picking it on the next try) why won’t you end up with 10 of each number?
It may be easier to see how this works with smaller numbers. You flip a coin twice. It’s either heads or tails, 50% probability, and the two flips are independent. Your question is “why don’t you end up with exactly 1 heads and 1 tails?”
Flip the coin once. Suppose (for the sake of argument) that it lands on heads. You flip it again. There is nothing to influence the coin to land tails here – it’s still 50-50 whether it lands heads or tails.
Lets make it more simple.
If we flip a coin twice, each side have 50% chance of being picked each time.
Do you understand why you might not end up with exactly one heads and one tails?
Because your brain (and my brain and everyone else’s brains) is really dumb about instinctively thinking about probability. We naturally expect things to even out but that’s only generally true and only for trying lots of times. Plus a bit of magical thinking that our knowledge changes how reality works.
The important thing in your example is that each time you pick a number, your process doesn’t know or care what the previous results were. If your first pick was a six, you’re right to think the next one won’t be a six but that’s only true because there’s only one result that’s a six and nine that are not–just like in the first pick. Getting a second six didn’t become less likely even though it feels like it should.
The same thing happens in the rest of your trials. At no point does the universe go “Uh oh, there haven’t been enough twos. Better squeeze a couple more in before the end.” In fact, the only way to guarantee exactly 10% of each number is to have an infinite number of picks.
Because randomness doesn’t care about what has already happened. You still have a 10% chance at each number every time. Even if you’ve hit that one a few times.
It’s called the Law of Small Numbers.
I just went and flipped a fair coin, and it landed on heads. Cool! Because it’s a fair coin, and the odds are 50/50 of heads or tails, I should get heads half the time and tails half the time, right?
But what would make it get tails next time? Does God Himself flip the coin in midair to make sure it ends up tails and stays fair? I just flipped it again and got heads, apparently not!
The short answer is: the odds of flipping a coin are _always_ 50/50, no matter what the last coin flip was or what the last 100 were.
But if I flip 100,000 coins, there’s a very high chance that just about 50,000 of them end up heads and 50,000 end up tails, give or take a few extra heads or tails. That’s called the Law of Large Numbers.
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