Written by Colin+ in ask uncle colin.
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).
I’m going to give the events letters:
We’re told:
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.
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