I have the following question:
Suppose we take a fair two-sided coin - the probability of getting a heads is the same as the probability of getting a tails.
We know that the result of the current flip does not influence the results of the next flip. We can use this fact to disprove the idea of the "hot hand" - that is, if we see a sequence of 100 consecutive heads, we know that the probability of the 101st flip also being heads is as likely as the 101st flip being tails. Therefore, betting on the 101st flip to be tails on the basis that the coin "probably ran out of luck and the streak is likely to be over" has no mathematical rationale.
However, I would like to approach this question from a different angle. Let's say that we assign heads as "1" and tails as "0". Suppose we flip a coin 10 times and 9 of them are heads - we take the sum of this sequence and get an answer of 9.
As such, the coin toss follows a Binomial Distribution (Bernoulii Trials). This means that if we assigned a Random Variable "X" to follow a Binomial Distribution ~ (n,p) ... we then could create a new Random Variable which would correspond to the sum of IID (Independent Identically Distributed) Binomial Variables, e.g. Y = X1 + X2 + ... + X10 .
Now, using the laws of Probability and Expected Values, we could likely find out the Expected Value of Y.
I am guessing that the sum of independent Binomial Variables is still a Binomial Distribution (Sum of two independent binomial variables) - this being said, if the probability of a success is 0.5, in "n" trials, the Random Variable "Y" is likely to assume certain values on average with higher probabilities compared to other values. This being said, Y having a value of 9 seems unlikely to me - and likely grounds to believe that this coin did not have a success probability equal to 0.5.
This being said, are the following conclusions true?
Given the probability of a success = 0.5 and events being uncorrelated with other events - a streak of 10 consecutive heads is as likely/unlikely as any other arbitrary sequence of coin flips - and the probability of the 11th flip being heads is also equally/unlikely to be heads.
However, the Expected Value of sequences originating from a "n" coin flips with a probability of success "p" - is likely to assume certain values more likely than other values. Thus, if I see 10 consecutive flips where the outcomes is heads - I now have reasons to believe that this coin likely did not originate from a Binomial Distribution with p = 0.5
Is this correct?
Thanks!
