In a hypothetical scenario,
if Lebron James made 10 successful shots in the same spot consecutively, what is the probability that his 11 shot is successful? Is it the same probability as the 1st? or did any of his prior shots affect his 11th throw?
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What about in the instance of coin toss, similar scenario,
if i throw head for 10 times, what is the probability that i will get another head in the 11th time?
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Thanks
In: Mathematics
There’s an easy mathematical answer and there’s a deeper answer.
The easy mathematical answer is it doesn’t matter. Both coin tosses and free throws are “independent events”. That means there isn’t anything that changes the next outcome based on the previous outcome. Let’s talk about free throws later, I already hear objections. For a coin toss, there’s nothing cosmic that ever makes one side more likely than the other. If you have flipped 99 heads the probability of the next coin being tails is still 50%. (I will have an interesting footnote about this.)
The deeper answer, and the objections I hear, is that some things aren’t really as “independent” as the math dictates.
If I try to make a free throw and fall short, I’m going to throw the ball harder next time. If I overshoot, I’ll use less force next time. If I make the first shot, I’ll try to do it the same way. So for an inexperienced person, to some extent we kind of expect if they take 4 or 5 shots the last few shots are the most likely to make it, with some falloff as they get tired.
But Lebron James is a professional who has practiced tens of thousands of free throws. He knows how to sink one and is making fewer adjustments between throws. If he has sunk 10 shots in a row without moving, all we can go by is that it must be very likely his 11th will make it. But if we look at his lifetime free throw percentage, we should use that instead. I did a quick search and it says he has a 73.5% rate. That doesn’t mean he’s never made 10 in a row. It means if he’s made 1,000 shots, 735 of them were successful. That’s way better than my percentage would be. So the problem here is, “How likely is a free throw to be made?” is very unique to each person. That’s why coin tosses are better for probability. 10 coins should behave the same way unless we know they are manufactured very differently.
That said, even coin tosses are weird. A recent paper proved that due to some interesting Physics, a coin is ever-so-slightly more likely to land in the same orientation it was when you flip it. So if you held it heads-upwards every single toss, they found that human coin tosses have a teeny-tiny bias, slightly less than 1%, towards heads. So if your life ever depends on a coin toss and you get to make the toss, be careful to bet on which side is up. If you call it in the air and couldn’t see how it was held, it sort of balances out.
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The only way this is not true is for “dependent” events, which are events where something about the previous attempts changes how future attempts work. A Bingo game is a good example. The game starts with balls 1 through 100 in a bin. If you take one ball out and it is 35, now there are only 99 balls. At the start, each ball only had a 1% chance of being selected. Now each ball has a 1 in 99 chance, which is slightly more than 1%. So if you are betting on one particular number being pulled, the odds it is pulled first are very low. But each number after that it gets more likely, and you will never pull all 100 balls without seeing your number unless someone cheated.
That is not so with coin tosses. Nothing in the universe prevents a coin from landing heads 1,000,000 times in a row. The key part to understand here is nothing “decides” how the coin lands. There is no cosmic force tracking every coin toss and trying to balance them. It is an act that can end in one of 2 ways, and ideally nothing about the last flip changes the coin, so it will always end one of those two ways with equal probability.
The hard part of this is if you’re gambling, if you choose the side that doesn’t land you think you “chose wrong”. What gamblers have to realize is “bad beats” exist. There have been poker plays where a person had 95% or better odds of winning and *lost*. That doesn’t mean they shouldn’t have bet on their own win. It just means they got unlucky. Being “good” at gambling means you know it’s smart to always bet on the highest odds of winning and to never let the times you were unlucky change what you know to be true about probability. People who are “good” at gambling understand it is possible to do everything “right” and lose. That’s why it’s called “gambling”.
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