Shortcuts With Chained Probabilities

February 17th, 2021
math
Let's say you're considering an activity with a risk of death of one in a million. If you do it twice, is your risk two in a million?

Technically, it's just under:

1 - (1 - 1/1,000,000)^2 = ~2/1,000,001
This is quite close! Approximating 1 - (1-p)^2 as p*2 was only off by 0.00005%.

On the other hand, say you roll a die twice looking for a 1:

1 - (1 - 1/6)^2 = ~31%
The approximation would have given:
1/6 * 2 = ~33%
Which is off by 8%. And if we flip a coin looking for a tails:
1/2 * 2 = 100%
Which is clearly wrong since you could get heads twice in a row.

It seems like this shortcut is better for small probabilities; why?

If something has probability p, then the chance of it happening at least once in two independent tries is:

1 - (1-p)^2
 = 1 - (1 - 2p + p^2)
 = 1 - 1 + 2p - p^2
 = 2p - p^2
If p is very small, then p^2 is negligible, and 2p is only a very slight overestimate. As it gets larger, however, skipping it becomes more of a problem.

This is the calculation that people do when adding micromorts: you can't die from the same thing multiple times, but your chance of death stays low enough that the inaccuracy of naively combining these probabilities is much smaller than the margin of error on our estimates.

Referenced in: Peekskill Lyme Incidence

Comment via: facebook, lesswrong, substack

Recent posts on blogs I like:

Elixir's Last Dance

On May 18th, the contra dance band Elixir had their last gig ever. The dance was packed: there were three hundred people. It was the only dance BIDA has ever done where they sold tickets. People flew from across the country just to hear Elixir play one la…

via Lily Wise's Blog Posts June 5, 2025

Body Language For Trans People

When I first came out as trans, resources for trans people were full of advice about body language.

via Thing of Things June 2, 2025

Workshop House case study

Lauren Hoffman interviewed me about Workshop House and wrote this post about a community I’m working on building in DC.

via Home April 30, 2025

more     (via openring)