N Heads in a Row

[petrw1]

Given a coin that lands heads with probability 0 < p < 1, what is the expected number of flips needed to get N heads in a row?

Solve with p = 0.73 and N = 20.

Then can someone give a general method / formula based only on p and N?

Thanks

[Nick]

It's the negative binomial distribution (with parameters N and p).

[petrw1]

Pardon my ignorance but ...

maybe I'm not understanding this right, but it looks like negative binomial distribution is used to determine, for instance, the expected number of tails before a pre-determined number of heads, but not necessarily in order

[Nick]

Originally Posted by petrw1

maybe I'm not understanding this right

I think I misunderstood you!

In that case, you can work it out from first principles.
Fix a positive integer \(k\geq N\).
How many ways are there of throwing \(k-N\) times without getting \(N\) heads in a row?
What is the probability of each of those possibilities followed by \(N\) consecutive heads?

[wblipp]

Originally Posted by Nick

How many ways are there of throwing \(k-N\) times without getting \(N\) heads in a row? What is the probability of each of those possibilities followed by \(N\) consecutive heads?

Not quite right, either. The \(k-N\) might have ended with one or more heads. I usually grab the density using the transition matrix for the Markov Chain of #heads in a row with an absorbing state at \(N\), but this question asks for the mean.

I'd try an inductive approach on the number of heads. To get one head in a row, the first toss is a head with probability p or a tail with probability (1-p). If heads, you are done. If tails, you need the to start over. So

X1 = p*1 + (1-p)*(X1+1)
X1 = 1/p

For k in a row, you need k-1 in a row, then one more toss either finishes it or starts it over.

Xk = p*(X(k-1)+1) + (1-p)(Xk+X(k-1)+1)
Xk = X(k-1)/p + 1/p

The logic of the last equation is conditioned on the results of the flip after first achieving k-1 heads.

Check that arithmetic - I fixed several errors before posting and might still have some left - but the logic is solid.

[science_man_88]

Originally Posted by petrw1

Given a coin that lands heads with probability 0 < p < 1, what is the expected number of flips needed to get N heads in a row?

Solve with p = 0.73 and N = 20.

Then can someone give a general method / formula based only on p and N?

Thanks

Take 1-(1-p)ⁿ to be the probability of getting heads n times ? Factor in how far you've advanced and find where it hits 50% ?

[wblipp]

Oops - this is Homework Help. I had fun working out the inductive equations and their general solution, but I suppose I shouldn't post all that here. So I stand by the previous suggestion that N+1 heads in a row takes N heads in a row plus one more, and then you may be done or you may need to start over. To check your work, I get 2001.6 for p = 0.73 and N = 20.