# Bone Marrow Odds

I don't remember doing it - although I'd meant to for some time - but apparently I signed up for the British Bone Marrow Registry. (If you're between 17 and 40, you can sign up the next time you give blood; the more people on the register, the more likely it is for people who need a transfer to find a match.)

But this post isn't about how lovely and generous I am1. Instead, it's about a stat in the letter they sent me: in any given month, there is a 1-in-240 chance of a given person on the register being matched with someone who needs a donation.

### Is that a big number?

For me, that's surprisingly high. In any given year, that's about 5%.

I worked that out as $\frac{1}{240} \times 12$, although that's only rough; better is to do $1-\br{\frac{239}{240}}^12 \approx 0.0489$ - meaning that 5% is a pretty good guess.

In any given decade, that's closer to 40% - we're into "$e$ approximation" territory now: since $\br{ 1 - \frac{1}{240}}^{240n} \approx e^-n$, it's clear that $\br{1-\frac{1}{240}}^{120} \approx e^{-\frac{1}{2}}$. Since $e^\frac{1}{2} \approx \frac{5}{3}$ (I know it's about 1.65), the probability of not being picked over 120 months is about $\frac{3}{5}$, so we're looking at a 40% match probability.

### Will I get to save a life?

I hope so - and I think the odds are good. I don't know if there's an upper age limit (maybe 65?), so let's say I have 25 years ahead of me (all being well). That's 300 months, which translates to an $e^{-1.2}$ chance of no match. I know that $\ln(3)=1.1$, so $e^{1.2}$ is about 10% more than that - let's call it $\frac{10}{3}$. That gives me a 30% chance of no match, or 70% chance of being called.

* Find out more about the bone marrow registry here.

## Colin

Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.

1. at least, not entirely []

### One comment on “Bone Marrow Odds”

• ##### Evelyn Lamb

Wow, that’s a lot higher than I would have thought as well, based in part on the fact that I signed up for the U.S. bone marrow registry when I was 17 or 18 and have never been called. (I’m…a bit older than 17 or 18 now.) But the U.S. and U.K. registries may have completely different odds.

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