Are the chances of x event occurring over course of a year and course of a day the same? I have a simple question I need answered to settle a debate with two concerned female friends.
Take this quote: "Of every 100 women whose partners use withdrawal, 4 will become pregnant each year if they always do it correctly."
Does this mean that the chance of becoming pregnant from any single instance of intercourse using withdrawal is also 4 of 100, 4%?
I think the answer is no. My friends say yes.
As a simple counter-example, the chance of flipping a coin and landing heads at least once over the course of two flips is 75%:
TT
TH
HT
HH
Whereas the chance of landing heads with a single flip is 50%. Clearly the chance of a heads in a single event is lower than the chance over x events (where x is greater than 1).
Who is right?
 A: So, if a random couple has sex $N$ times a year, and the probability each time that the woman gets pregnant is $p$, then the probability that she won't get pregnant is $(1-p)^N$. So if the probability that she gets pregnant in a year is 4%, then the probability not is $0.96$, and $p=1-\sqrt[N]{0.96}$.
It's more complicated if $N$ itself is a random variable - assuming uniformity in the space makes it somewhat easier.
For example, if 4% have sex every day, and the rest have sex 39 times a year, the couples average 52 sex acts per couple, but the value of $p$ changes, because those 4% are very much more likely to get pregnant.
So using this statistic really only gives you some vague area of estimating the probability of pregnancy in any individual sex act.
Ultimately, each paper might mean a different thing here, and you'd have to read more details to find out what they actually meant. But they should never mean that each sex act has a 4% chance of pregnancy, unless they are specifically dealing with a sample where couples only have sex once a year.
A: I think the intent is that if they live with their partners regularly during the year, then 4 of 100 will become pregnant.
In other words, let's say regular life means 1 times a week, with approximately 52 weeks a year. Then, of 1 woman participates in $52$ acts a year, so 4 out of 100 is
$$
\frac{4}{100 \cdot 52} = \frac{1}{1300} \approx 0.0769 \%,
$$
and that is probability of getting pregnant in one single act.
