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I have a faked coin and I throw it until I first get a head H.

$X=1$ for H with prob. $p$

$X=0$ for T with prob. $1-p$

I am a little bit confused right now with two things:

What is the probability that the experiment (after finite time) will end?

Let $N$ be the number of tails until the first time head appears. What is the probability distribution for the random variable N ?

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try reading this – jerad Mar 7 '13 at 23:18
up vote 1 down vote accepted

Edit: Sorry I wrote it wrong: A and B independent implies $P(A\cap B)=P(A).P(B)$ (not $P(A|B)=P(A).P(B)$). but that exactly what you wanted.

First notice that each coins flip are independent hence $$P(X_1=1\cap X_{n-1}\cap...\cap X_1)=P(X_n=1)*P(X_{n-1}\cap ...\cap X_1)$$ hence $$P(X_n=1\cap X_1=0\cap X_2=0,...)=P(X_n=1)*P(X_1=0)*P(X_2=0)*...$$ I think this answer your question for N.

As for the probability the the experiment end after a finite time, its the probability that it finish after one step plus the probability it finish after 2 step plus etc. henc $$\sum_{n=0}^{\infty}P(N=n)$$

Hope this helps.

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Thanks, one more question. If I take for exmaple B={we throw at least one time a T=tail}. What is this event in terms of the random variable N ? – Voyage Mar 7 '13 at 23:36
If you mean the probability to have at least one tail over n trial, then its 1 minus the probability to have n heads over n trial. Otherwise i didn't understood what you mean :s – wece Mar 7 '13 at 23:49

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