what is the expecxted length of the initial run for heads?

Asked Jan 07 '23 at 07:59

Active Jan 07 '23 at 07:59

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Suppose a coin is tossed repeatedly with a probability of head appears in any toss being $p,~0<p<1$.

I want to find the expected length of the initial run of heads.

Here initial run for heads means the number of heads until the first tail appears.

So if $p$ is the probability of head, then $1-p$ is the probability of tail at any toss.

The outcome can be

at 1st toss $\{H\}$

at 2nd toss $\{HH,HT,TH,TT\}$

at 3rd toss $\{HHH,HTT,HHT,TTT,THH,TTH,THT,HTH\}$

$...$

The distribution function seems to be $m(r)=(1-p)^{r-1}p$ at $rth$ toss.

But I am unable to find the expected length.

I seek help.

asked Jan 07 '23 at 07:59

MAS

The answers to this question should help you: Expected number of tosses till first head comes up – Lynn Jan 07 '23 at 08:12
@Lynn, thanks. The probability of 1st succes is as mentioned in my question $P(X=r)=(1-p)^{r-1}p$ and its expectation or expected lenth is $E(X)=1/p=2$. Am I right ? – MAS Jan 07 '23 at 08:17

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