Ask Uncle Colin: Missing Planes

Dear Uncle Colin,

I’m told that 70% of the aircraft that go missing in a certain country are subsequently rediscovered. Of those that are recovered, 60% have an emergency locator, and 90% of those that aren’t recovered, don’t have a locator. Supposing an aircraft has disappeared, what’s the probability it has a locator?

Likelihood Of Search Technology

Hi, LOST, and thanks for your message!

As is often the way, there are several approaches here. I’ll start with the one you’ll most usually see (applying Bayes’s Rule formulaically) and then show you my preferred way (a table).

Bayes

I’m going to give the events letters:

  • The event that a missing plane is recovered is $R$
  • The event that a missing plane has a locator is $L$

We’re told:

  • $P(R) = 0.7$
  • $P(L | R) = 0.6$
  • $P(L’ | R’) = 0.9$

And we need to work out $P(L)$.

From the second equation, we know $\frac{P(L \cap R)}{P(R)} = 0.6$, and substituting in the value of $P(R)$ tells us $P(L \cap R) = 0.42$.
I don’t much like that third equation, and I’d prefer to rephrase it: 10% of the unrecovered planes have a locator, so $P(L | R’) = 0.1$.

That means $\frac{P(L \cap R’)}{P(R’)} = 0.1$, and substituting in $P(R’) = 0.3$ gives $P(L \cap R’) = 0.03$.

Either a plane has a locator or it doesn’t, so $P(L \cap R) + P(L \cap R’) = P(L)$, which works out to be 0.45.

A Table

I start with a table like this:

  R R' Total
L
L'
Total 70 100

The only immediate information we have is that for every 100 planes that go missing, 70 are rediscovered - although we can immediately fill in “30” in the total / R’ box.

We also know that 60% of the 70 recovered planes - 42 altogether - have a locator and the 90% of the 30 unrecovered planes - making 27 - don’t. So we can fill in a bit more:

  R R' Total
L 42
L' 27
Total 70 30 100

And now we have enough to fill out the table! L/R’ must clearly be 3 to make the totals work; similarly, L’/R must be 28.

  R R' Total
L 42 3
L' 28 27
Total 70 30 100

Then we can fill out the last column by adding up across the way (and checking that it adds to 100):

  R R' Total
L 42 3 45
L' 28 27 55
Total 70 30 100

Then you can just read off: out of every 100 lost planes, a total of 45 have a locator.


I have to say, that seems to be an awful lot of planes to be going missing! But in any case, I hope that helps.

- Uncle Colin

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.

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