Like. I know they are different and that one is less likely. But could someone explain this in a way that I can explain it to my partner? I know it is true but cannot remember anything about why and how to explain my point.
In: 6
We often assume (usually pretty reasonably) that events are “Independent”. Probability only really works in the simple, easy-to-understand way that is taught in 300-level undergrad intrductory courses if you assume this. You roll a die, it doesn’t remember what just happened and change its next roll because of it- the die has no memory. It is, after all, an inanimate object.
If you roll six dice, each number has six different one in six chances to come up. But the dice can’t see each other! And even if they could, they can’t change their roll. Thus, if you roll six dice, it is possible not to get one of each number. Quite common, in fact.
Many repeated chances gives many possible outcomes. If you take 10 7% chances, it is possible (if quite unlikely) to hit the chance all 10 times- something that *cannot* happen with one 70% chance.
Possibly confusing the matter is “Expected Value”. With expected value, 10 7% chances are indeed the same as one 70% chance. This is because their outcomes are the same *in the long run*. 10 7% chances will sometimes turn out quite differently than one 70% chance, but one million 70% chances and ten million 7% chances will approximately always look very, very similar.
But life is lived in the small moments. Thus does probability occasionally make fools of us all.
Because each chance is separate, but we tend to think of “ten 7% chances” as combining together into 1 70% chance. But this is just flat wrong. Each chance is unique and none is affected by the other. So they don’t “add up” to 70%, rather they remain ten separate chances.
It’s like when someone’s flipped a (non-trick) coin 5 times in a row and they’ve all come up heads, so they think tails is “due”. The coin doesn’t care about or know how many times it’s been flipped and what the results were. It’s just a coin, and every time you flip it there’s a 50/50 result.
Look at it from the perspective of just percentages.
Say “winning” is having more pizza than anyone else.
If you have a single pizza and cut it into 10 slices, you can be given 7 slices and have 70% of the pizza. You win.
Now if you have 10 pizzas, each cut into 10 pieces, and are given 7% of each pizza, you’ll have less than one slice of each pizza. The total of all those smaller slices compared to the rest of the pizza you don’t get is a small share compared to the last example of a single pizza. There’s a much higher probability you don’t win.
Suppose you have 100 boxes, either 70 or 7 boxes are filled with prizes. One opportunity with 70 boxes filled gives you less opportunity to mess up than picking many times with only 7 filled.
Think of it as a game where a player has to correctly guess a number.
(70%) Player 1 has to guess a number between 1-100 and there are 70 correct answers.
(7%) If player 2 has guess a number between 1-100 and there are 7 correct answers.
Player 1 has one chance to guess a 70 numbers out of 100.
Player 2 has ten chances to guess 7 numbers out of 100.
Even with 10 times more chances, it is still harder to guess just 7 numbers.
It gets easier to understand the chance you increase the range of numbers you have to guess (e.g. 700/1000, 7000/10000)
Change the numbers a bit to keep it simple. Flip a coin, 50% chance to get heads up on one toss. If you do that twice, you have the following equal possibilities:
* Heads+Heads
* Heads+Tails
* Tails+Heads
* Tails+Tails
So, 3 out of 4, or 75% chance to get one heads up. Which is different from a 2×50%=100% chance.
Can we tweak the numbers a bit? Would you rather have a single 100% chance, or 100 1% chances to win $1 million?
Let me give you an example: There are two boxes A and B. Box A has 7 Good Apples and 3 Bad ones and Box B has 1 Good Apple and 9 Bad ones. The probability of someone picking a good apple at random from Box A and Box B is .7 and .1. Also, note that every time an Apple is picked the Apple is put back into the same box (the probability accounts for the setting with replacement). So, no matter how much ever time a person tries to pick from box B the probability of picking a good Apple is 0.1 as every pick is independent.
So, if we say increase the number of apples one can pick from a box. The worst case is that a person has to pick 10 apples from box B to have one good Apple, which comes to 4 in Box A.
**The real ELI5 is because they are isolated events.**
Slightly more detailed example…
If you have a bag with 100 marbles (7 blue, 93 red), you have a 7% chance of picking a blue marble. If you put the marble back into the bag and try again, you still have a 7% chance of picking a blue marble the second time. The second marble pick is an isolated event, not affected by the previous attempt. So the probability of picking a blue marble is 7% each time. If you try 10 times, the probability ‘resets’ each time, they don’t ‘stack’.
However, if you don’t put the marble back into the bag, then the attempts are not isolated events and the probability of picking a blue marble changes based on the results of the previous attempt.
Probability doesn’t add, it multiplies. Think about it, you could take 20 7% chances and still fail, so it can’t simply add up, but the chance of all fails gets smaller into near nothingness as more tries are included. To get the probability of ten 7%s, take the odds of it not happening (0.93), multiply it by itself ten times (0.93^10). That equals 0.48, the odds of failing all after seven tries. Take the inverse (1-0.48=0.52) and you have a 52 percent chance of success from 10 7% s.
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