Probability of past instance

Question

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

Answer 1

You are invoking something called the “Gambler’s Fallacy”, which holds that probability sort of gets stored up and that the longer an event goes without occurring, the more likely it becomes.

This is called a fallacy for a reason.

While there are random events whose probabilities do depend on previous events, that’s definitely not the case in a coin toss, and perhaps not the case for LeBron.

No matter how many heads you get in a row, the odds of the next heads will **always** be 50%. Yes, the odds of you have an event where you get 11 heads in a row is pretty unlikely (about 1 in 2000), but the coin has no memory. The coin doesn’t think about the fact that it’s only been flipping heads for a while, and isn’t that strange.

With LeBron James, if we say that each of his shots makes it with some probability that’s related to his inherent skill and traits and not the state of the game, that’s also true. In practice, other stuff can affect it. He can get tired. A minor injury he took early in the game could be affecting him, or he could be getting complacent, or maybe he’s trying to break a previous record. Humans are complicated systems^[citation ^needed] and it’s hard to isolate them from all influences.

But I think the spirit of your question is that, no, the past shots do not affect the current one.

If you want, you can play a game to convince yourself of this at home: flip a coin a whole bunch of times, and while you likely won’t get to 10 heads in a row, if you keep a record of each flip, you can easily show that the odds of heads or tails after 3 heads in a row are exactly the same as after 2 heads, or after 2 tails, or after 4 tails, or after heads-tails-heads, or whatever.