How do I find the expected value of trials to achieve an amount of successes in a row?

Asked May 02 '18 at 05:31

Active May 02 '18 at 05:31

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So, I'm creating a probability simulator, and my goal is to compute how many trials of an experiment with $n$ fair outcomes would it take to achieve $x$ successes $y$ times.

For example, say I do an experiment with $2$ fair outcomes (flipping a fair coin), and I flip until I see a pattern such as $HHH$ come out from it, $3$ times, with no limit of how many flips I do total or how many failures I have.

What equation would I use to calculate an expected value and variance of that value from that?

I tried it myself, and, calling the probability of heads $p$, I got the equation $E[x]=n^{lr}$, where $l$ is pattern length, and $r$ is experiment repetitions. I know, or at least, feel like this is incorrect.

asked May 02 '18 at 05:31

DDPWNAGE

It seems that $p$ should be somewhere in $E(x).$ Should $p=1/n$? – coffeemath May 02 '18 at 05:36
1

See this related question where it is asking about the specific result of five heads in a row on a fair coin and try to generalize from there. – JMoravitz May 02 '18 at 05:38
@coffeemath I feel like it should be, too. $n$ is the amount of outcomes. My thinking was, $p = 1/n$. I read somewhere about using $k/p$ too, where $k$ was successes. – DDPWNAGE May 02 '18 at 05:39
What do you means by "achieve $x$ successes $y$ times"? In your example, $n=2$, but does the "$3$ times" refer to $x$ or $y$? – Math1000 May 02 '18 at 07:48
@Math1000 $y=3$. – DDPWNAGE May 02 '18 at 07:51

How do I find the expected value of trials to achieve an amount of successes in a row?

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