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Could you tell me if my reasoning is right? I'm new in probability so I'm still not secure of how I should use the classical distributions to model problems.

My problem is the following: I have a success-failure experiment such that the probability of success is $p\in(0,1)$ ($p$ is supposed to be very little, but this shouldn't change the reasoning). If we run the experiment one time, the probability of success is $p$. If we run the experiment $n$ times, the probability of having $k$ successes is $\binom{n}{k}p^k(1-p)^{n-k}$, for $0\leq k\leq n$. This is modeled by the binomial distribution.

What I want to know is an inverse problem, we can say. How many experiments we should run to have one success with probability at least $50\%$? I'm looking for the minimal number $n$ of experiments.

My idea: We can keep $n$ unknown and ask for the probability of having at least one success in $n$ experiments. This is given by $\sum_{k=1}^n \binom{n}{k}p^k(1-p)^{n-k}$. It's possible to use some program to find the minimum $n$ such that $\sum_{k=1}^n \binom{n}{k}p^k(1-p)^{n-k}\geq \frac{1}{2}$. In fact, I did this for some values of $p$, in particular, for $p=\frac{1}{10000}$ we have $n=6932$.

But I'm not sure of this argument. I was looking for the number of experiments necessary to have one success with probability at least $50\%$, but ended up calculating the number of experiments necessary to have at least one success with probability at least $50\%$.

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2 Answers 2

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The number of successes in $n$ independent Bernoulli trials with probability of success $p$ is a binomial random variable $X$, such that $$\Pr[X = k] = \binom{n}{k} p^k (1-p)^{n-k}, \quad k = 0, 1, 2, \ldots, n.$$ Therefore, the probability of observing at least one success in one trial is the complement of the probability of observing no successes: $$\Pr[X \ge 1] = 1 - \Pr[X = 0] = 1 - \binom{n}{0} p^0 (1-p)^{n-0} = 1 - (1-p)^n.$$ We then wish to find the minimum $n$ such that $$0.5 \le \Pr[X \ge 1] = 1 - (1-p)^n;$$ that is to say, $$n \ge -\frac{\log 2}{\log(1-p)} = \left\lceil -\frac{\log 2}{\log (1-p)} \right\rceil.$$ For $p = 10^{-4}$, this gives $n = 6932$ as you wrote.


If, however, you want the probability of exactly one success to be at least 50%, this would be $$0.5 \le \Pr[X = 1] = \binom{n}{1} p^1 (1-p)^{n-1} = np(1-p)^{n-1}.$$ The solution of this inequality does not have an elementary closed form for general $p \in (0,1)$ and positive integer $n$. However, we can show that there is no solution to this inequality if $p < 0.44303\ldots$, since for a fixed $p \in (0,1)$, treating $n$ temporarily as a continuous positive real and differentiating $\Pr[X = 1]$ with respect to $n$, we can see that a local extremum occurs when $$n = -1/\log(1-p),$$ attaining $$\Pr[X = 1]_{\text{max}} \approx \frac{-p}{e(1-p)\log(1-p)}.$$ And this expression is less than $1/2$ for $p < 0.44303\ldots$. Of course, $n$ is a positive integer, so this is just an approximation, but it remains the case that when $p$ is "small," no number of experiments will guarantee that the probability of exactly one success will exceed $0.5$.

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  • $\begingroup$ You can see my problem in this lines: suppose you have a die with 10000 sides numbered and pick one specfic side. How many times you should throw the die until you have at least $50\%$ of chance of coming up this side? I got a little confused about what to use: at least one success or exactly one success. $\endgroup$
    – diff_math
    Sep 17, 2015 at 23:51
  • $\begingroup$ My interpretation of the wording above is "at least one." $\endgroup$ Sep 17, 2015 at 23:53
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    $\begingroup$ You can close the gap between your approximation $0.443$ and the true bound of $0.5$ by considering $n=1,2,3$ $\endgroup$
    – Henry
    Sep 17, 2015 at 23:54
  • $\begingroup$ @diff_math That question is a completely different one to the binomial question! You are using a binomial model when your actual model is geometric. The fundamental difference is that in a binomial model, the number of trials is fixed in advance. In your proposed experiment, you continue to conduct trials until you achieve success. $\endgroup$
    – heropup
    Sep 17, 2015 at 23:54
  • $\begingroup$ @Henry Indeed, I agree. I just didn't bother going through those last steps since the range of $p$ of interest is so much smaller. $\endgroup$
    – heropup
    Sep 17, 2015 at 23:55
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The number of experiments necessary to have at least one success with probability at least $50\%$ can be calculated as the number of experiments necessary to have no success with probability at most $50\%$. This requires $(1-p)^n \le \frac12$ so $n \ge \dfrac{\log 2}{- \log(1-p)}.$ With your example of $p=\frac{1}{10000}$ this gives $n\ge 6931.1$ (approximately, and you will round up to the next integer).

The number of experiments necessary to have exactly one success with probability at least $50\%$ requires $n p(1-p)^{n-1} \ge \frac{1}{2}$ and this will not have a closed form solution if you do not use something like Lambert-W functions. But for small $p$ there will be no sensible solution to this anyway: in your example of $p=\frac{1}{10000}$, the maximum probability of exactly one success happens when $n=9999$ or $10000$ and is then about $36.78978\% \approx \exp(-1)$ which is too low to answer your question.

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  • $\begingroup$ Are you saying that $\sum_{k=1}^n \binom{n}{k}p^k(1-p)^{n-k} = 1-(1-p)^n$ ? $\endgroup$
    – diff_math
    Sep 17, 2015 at 23:46
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    $\begingroup$ Yes, I am, or was. I deleted my answer because I had written a solution on the assumption you meant at least one success. It is often impossible to have a probability $\gt 50\%$ of exactly one success. Basically, if we have a large enough sample to have a good chance of at least one success, chances are good we will have $2$ or more. $\endgroup$ Sep 17, 2015 at 23:47
  • $\begingroup$ @diff_math Yes. It obviously is. Summing from k=0 to n is the probability of anything from no to all successes; which is $1$. So if we subtract the probability of no success from $1$ we get the required probability of at least one success. $\endgroup$ Sep 17, 2015 at 23:49
  • $\begingroup$ It seems that when I say there is no solution for small $p$, I mean $p \lt \frac12$. When $p=\frac12$ then $n=1$ or $2$ are solutions and when $p \gt \frac12$ then $n=1$ is a solution to your "exactly one" question $\endgroup$
    – Henry
    Sep 17, 2015 at 23:49
  • $\begingroup$ Thanks for clarifying. I will rewrite my comment on the other answer, because this is what made me confused. You can see my problem in this lines: suppose you have a die with 10000 sides numbered and pick one specfic side. How many times you should throw the die until you have at least $50\%$ of chance of coming up this side? I got a little confused about what to use: at least one success or exactly one success. $\endgroup$
    – diff_math
    Sep 17, 2015 at 23:54

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