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

Consider flipping a coin three times, there are eight possible outcomes: TTT, TTH, THT, THH, HTT, HTH, HHT, HHH.

If you look at any individual coin flip you see that Tails occurs as often as Heads. T** happens four times, and H** happens four times; * T * happens four times, and * H * happens four times; ** T happens four times, and ** H happens four times. So clearly, any individual coin toss has 4/8=50% odds of being Heads or Tails.

But what about the odds of getting HHH? Well, that only happens 1/8=12.5% of the time.

Now what about the odds of getting HHH if we already have HH*? Well, if we already have HH* then that means that we are in a world where TT*, HT*, and TH* didn’t happen. So, despite those accounting for 6/8 of the total possible “three coin flips” outcomes, we’re only looking at what fraction of the time HH* (which is the remaining 2/8 possible outcomes) becomes HHH (1/8 outcomes). And what fraction of 2/8 is 1/8? Half. 50%.

Your hypothetical is *similar* in that each shot would be *independent* just as the coin flips were independent – so the odds of hitting the last shot is the same as the odds of hitting the first shot. However, your hypothetical will be different because some more complicated math is needed when the odds of each shot aren’t *exactly* 50:50. (Basically, if the odds are 75% he makes it, you’d have to *weight* out the possibilities differently; the first shot’s outcomes would look more like like (H,H,H,T)=(3H,1T) since that gives the proper 1/4 odds of hitting tails – then the second shots outcomes would compound on top of that like (H(H,H,H,T),H(H,H,H,T),H(H,H,H,T),T(H,H,H,T)) which would give (9HH,3HT,3TH,1TT) and so on.)