How can you approximate the number of trials to success given a particular Pr(Success)?

Question

I'm uncertain whether I should be able to intuit the answer to my question from a question that has already been asked but I can't, so I am asking the question anyway. Thus, I am looking for a clear easy to understand answer. A recent newspaper article reported that on average couples were able to conceive a child after 104 reproductive acts. Assuming indpendant binomial trials, that means for each act there was a 1/104 probability of success. I can do a quick simulation to show myself what the quantiles for this distribution look like, e.g. in R:

NSIM <- 10000
trialsuntilsuccess <- function(N=10000,Pr=1/104)
{
    return(min(seq(1:N)[rbinom(N,1,Pr)==1]))
}
res <- rep(NA,NSIM)
for (i in 1:NSIM)
{
    res[i] <- trialsuntilsuccess()
    if (i %% 10  == 0) {cat(i,"\r"); flush.console()}
}
quantile(res,c(.025,.975))

But it seems like there should be some simple equation or approximation that could be applied, perhaps a probit or poisson? Any advice on how to get the quantiles without running a simulation? Bonus points for providing a way to do the relevant calculations in R.

Answer 1

If I understand your question correctly you want to compute the quantiles for the "No of failures before the first success" given that $p=\frac{1}{104}$.

The distribution you should be looking at is the negative binomial distribution. The wiki discusses the negative binomial as:

In probability theory and statistics, the negative binomial distribution is a discrete probability distribution of the number of successes in a sequence of Bernoulli trials before a specified (non-random) number r of failures occurs.

Just invert the interpretation of success and failures with a setting of r=1 would accomplish what you want. The distribution with r=1 is also called the geometric distribution.

You could then use the discrete distribution to compute the quantiles.

PS: I do not know R.