Flipping Heads Consecutively

Aquafina · #1 $Old$ November 2nd, 2009, 06:50 AM

What's the probability of flipping n consecutive heads on a fair coin? What about an even number of consecutive heads?

For the first one, I simply got (1/2)^n, is this correct?

How will I solve the second? Any hints? There could be infinite number of arrangements as there are infinite even numbers

Grandad · #2 $Old$ November 3rd, 2009, 09:33 AM

Hello Aquafina

Quote:

Originally Posted by Aquafina $View Post$

What's the probability of flipping n consecutive heads on a fair coin? What about an even number of consecutive heads?

For the first one, I simply got (1/2)^n, is this correct?

How will I solve the second? Any hints? There could be infinite number of arrangements as there are infinite even numbers

Yes, your first answer is correct.

For the second question, I think the question means this:

A coin is repeatedly flipped until a tail occurs. What is the probability that there was an even number of heads before the tail?

So we need to work out the probability of 0 heads followed by a tail, 2 heads followed by a tail, 4 heads followed by a tail, ... and so on. Then add these probabilities together. I'll start you off:

$p(T) = \tfrac12$

$p(HHT) = (\tfrac12)^3$

$p(HHHHT) = ...$ ?

... etc

Then add these together and sum to infinity. (I make the answer $\tfrac23$ .)

Grandad