Not the probability calculator we deserve, but the one we need

One of the more useful tools on the internet

Photo by Sumudu Mohottige on Unsplash

I came across a tweet recently that linked to this video of a podcast by Andrew Huberman. The entire podcast is over four hours long (!), but the relevant part for this discussion is only a minute long and starts at 2:04:28. Transcript below:

So to make this very simple, all we need to know is that for women 30 years old or younger, because the probability of getting pregnant on any one attempt to conceive is 20%, well, then if that doesn't occur the first time, then she should simply repeat that at least five and probably six times before deciding to go to an OB/GYN and conclude that there's something going on either with the egg or, of course, it could be with the sperm because 20 times 5 is 100. So we're talking about cumulative percent-- so 20, 40, 60, 80, 100. And the six month there would take you to 120%, which is a different thing altogether. But in general, that's why OB/GYNs will tell their female patients, look, if you're setting out to conceive, try for about five or six months, and if you're not successful, come back and see me.

Oh no. 

At this point I had never heard of Andrew Huberman. Should have I? He has a very impressive social media empire. 1.3 million followers on Twitter. 5.43 million on YouTube. Six million on Instagram. I’d be thrilled if far less than one percent of his following ever reads EconSoapbox. He has a PhD from UC Davis and is a Professor of neurobiology at Stanford, so he is a very smart guy. 

That’s why this error is so bad. 

What he’s talking about is the odds of a woman getting pregnant. According to one estimate, a woman around 30 years old has about a 20 percent chance of getting pregnant every month. Now, this isn’t every woman. Some women are going to be unusually fertile and will have a higher probability, while other women are infertile and have a zero percent chance. That said, apparently the typical woman will have a twenty percent chance. I have no idea what the quality of research is behind that statistic but let's assume it’s accurate. The question then becomes, if a woman has a 20 percent chance of getting pregnant every month, at what point should she be concerned that she (or her partner) may have fertility issues?

According to Huberman, it’s after about five months. Why? Because to find the cumulative, five-month probability, one needs to add the individual month probability. Thus, if the odds of getting pregnant after one month is 20 percent, the odds after two months is 40 percent, after three months is 60 percent, after four months is 80 percent, and after five months is 100 percent. Therefore, something must be wrong after five months if a woman isn’t pregnant, because she hit 100 percent probability. 

This is not how probability works. 

To determine the cumulative probability of independent events, one does not add the probability of each attempt. Warning: Math ahead! Only for a minute so stay with me. Below is the correct equation: 

\( 1-(1-p)^{n} \)

Where p is the probability of the event and n is the number of attempts. Thus, if there is a 20 percent chance of success in any month and a woman is trying to get pregnant for six months, the equation becomes:

\(1-(1-.20)^{6}\approx.738\)

Or about 74 percent. The intuition of the equation is fairly straightforward. If the odds of success are 20 percent, then the odds of failure are 80 percent. That’s the (1-.20). If the odds of failure in one month are 80 percent, the odds of failing six times in a row are just .80*.80*.80*.80*.80*.80=.262 or roughly 26 percent. If the odds of failing six times in a row is 26 percent, then the odds of not failing six times in a row must be 1-.26= .74 or 74 percent. Simple. 

Adding the probabilities is wrong. Several minutes before the above clip, he even introduces independent probability by using a flipped coin as an example. Each coin flip is an independent event. So even if a flipped coin came up heads four times in a row, it doesn’t make heads less likely the fifth time. The coin doesn’t have a memory; every time a coin is flipped it’s a 50/50 chance. Just adding the probabilities is wrong, otherwise flipping a coin twice would result in a .50+.50=1.00 or 100% chance that heads (or tails) are flipped at least once. In reality, the odds of a coin coming up heads at least once after two flips is 75 percent. The odds of flipping zero heads after two attempts is 25 percent. 

Huberman has made a massive math mistake. This type of problem is taught to high school students in AP stats. A tenured STEM professor at Stanford should know better. On the other hand, everyone makes mistakes like this. I definitely have. I’m afraid of going into the weeds with how probability works not just because I don’t want to bore readers, but because I’ll likely make an error. Again, I’m not familiar with Huberman at all and I don’t want to be needlessly harsh. It’s clear he realizes something is amiss with his logic when he talks about something having a 120 percent chance of occurring. This is obviously impossible. Also, someone who churns out four-hour-long podcasts is going to make the occasional error. To Huberman’s credit, he added an explanation of the error in the YouTube video’s description and has the correct math. I think it’s fair to say this was a bone-headed error that has since been corrected. 

Regardless, this is a perfect opportunity to bring up a useful internet tool: the binominal probability calculator. I use this calculator every week. Now that might say more about me than anything, but everyone will find it useful sooner or later. The binominal probability calculator calculates the chances of a number of successes (or failures) given a series of independent events. This might sound esoteric, but it comes up a lot. 

Say your favorite basketball team is down by one point and a player on your team is taking two free throws. That player is shooting 71 percent from the free-throw line over the season. What are the odds he makes at least one free throw? Plug it into the calculator. In this case, p=.71, number of trials=2, and number of successes=1. The cumulative probability of (X≥1)=.9159. So there is a 91.6 percent chance the player makes at least one free throw. That also implies there is an 8.4 percent chance he misses both.

Say you’re playing Settlers of Catan and you’re about to win the game. The only thing that can prevent you from winning is if an eight is rolled in at least two of the next four rolls, which will allow the opponent to your right to win. What are the odds of that? Well, the odds of rolling an eight with two dice is 5/36 or .138. Thus the odds of rolling an eight at least twice in four rolls can be entered into the calculator. The probability is .138, number of trials is four, and number of successes is 2. The result is that there is a 9.4 percent your opponent will get lucky and beat you. 

See? The applications of binomial probability are practically endless!

Of course there are a few caveats here. First, binominal probability is only accurate if events are independent of one another. A basketball player might be shooting 71 percent from the charity stripe throughout a season, but that doesn’t necessarily mean they have a 71 percent chance of making a free throw when they are down by only one point at the end of an important game. Opposing fans distracting the player from behind the hoop might reduce those odds. If the player misses the first shot long, he might then be able to accurately compensate on the second shot. So the odds of success might not be 71 percent and the events might not be independent. 

Same with pregnancy. The typical 30-year-old woman might have a 20 percent chance of getting pregnant in any one month, but not all women are the same. Still, it's useful to know that if a typical woman has a 74 percent chance of getting pregnant after six months, there is a 26 percent chance she won’t. Now a 26 percent chance is unusual, but not rare. It’s about the odds of correctly picking the suit of a randomly drawn card. Things that have a 26 percent chance of happening occur all the time. So while it makes sense to see an OBGYN after six months of failing to get pregnant, it could just be bad, but not terrible, luck.

Next time you need to determine the odds of an event, use the binomial probability calculator. Whether calculating pregnancy odds, watching sports, or playing board games, it comes in handy.