2

Suppose you are flipping a coin whose probability of heads is $\frac{i}{j}$ where $i$ is the number of heads up to that toss and $j$ is the total number of tosses. Assume the coin lands tails the first time and heads the second. Thus the probability of heads on the third toss is $\frac{1}{2}$. What is the probability of getting exactly $35$ heads in 50 tosses?

My first thought was to use a binomial RV, but then you have a variable chance of success after each toss. You could bash this out with a computer since there are a finite amount of cases. My best thought is there is some sort of recursive solution, but I can't see what to do. Is there a known distribution this follows?

user2661923
  • 42,303
  • 3
  • 21
  • 46
Sebastian1213
  • 109

1 Answers1

6

Presumably you mean something like "exactly $35$ heads after $50$ tosses", since if you carry on tossing you will eventually get $35$ heads (with probability $1$).

Anyway, this is functionally equivalent to a Pólya urn. Try to prove by induction that after $n$ tosses the number of heads is equally likely to be any number between $1$ and $n-1$.

Especially Lime
  • 46,692
  • Wikipedia says the probability of exactly (x + n_1) white balls (heads) and (y + n_2) black balls (tails) is: $\binom{n}{n_1}\frac{x^{\bar n_1}y^{\bar n_2}}{(x+y)^{\bar n}}$. We have x = y = 1 since we have 1 of each to start. We do a total of n = 48 flips after this. We need 34 heads, so $n_1 = 34$. Plugging these in gives: $\binom{48}{34}\frac{1}{(2)^{\bar {48}}}$. Is this correct or am I misinterpreting since this number is near zero? – Sebastian1213 Aug 14 '24 at 14:49
  • 2
    That Wikipedia article is impressively obfuscated. It would be really helpful if it started with the elegant results which hold in the most simple and easy case. When $x=y=1$ (which is your case), the distribution after $n$ tosses is uniform on ${1,2,\dots,n-1}$, as EspeciallyLime says. (The term $1^{\bar{n}_1}$ works out to be $n_1!$, not $1$). – James Martin Aug 14 '24 at 15:07
  • I am struggling on the application of the inductive hypothesis. I see the probabilities for the n+1 flip given we had k heads in the first n flips, but I don't see how I can do this generally. – Sebastian1213 Aug 14 '24 at 23:09
  • @ Especially Lime: Great answer! I recently posted a question on Polya Urn - can you please check it out if you have time? https://math.stackexchange.com/questions/4953261/relationship-between-martingales-and-picking-balls-from-a-hat thank you! – konofoso Aug 15 '24 at 13:56