You want to go out for a walk this afternoon, but you're worried that it might rain. You turn on the television: the forecast is for rain. Should you give up on your walk?
You decide to do a little research. You go to the weather forecaster's website and discover that they claim a 90% accuracy rate: out of 100 days on which it rained, they predicted it would rain on 90 of those days. Sounds pretty good.
Digging a little deeper you discover that out of 100 days on which it did not rain, they correctly predicted it would be dry on 80 of those days. That's not too bad, either.
It looks like the forecaster is pretty reliable. You decide to go ahead with your walk but you take an umbrella with you.
It's bright sunshine the whole time! You didn't need the umbrella at all!
Why?
Because you didn't use Bayes theorem.
You see, it turns out that it rains only 10% of the time where you live. So in 100 days, it rains on 10 of those days. And the weather forecaster, with its 90% accuracy rate, would correctly predict rain on 9 of those 10 days.
However, it doesn't rain on 90 out of 100 days. But the weather forecaster would wrongly predict that it would rain on 20% of these. So on 18 days the forecast would be for rain when it didn't actually rain.
In total then, the weather forecaster predicts rain on 9 + 18 = 27 days out of 100. But on only 9 of those days does it actually rain. So the proportion of days on which it rains when the weather forecaster has predicted rain is 9/27, which is only one third. That's pretty unreliable.
The impressive statistic ("90% accuracy!") on the weather forecaster's website was the answer to the following question: "Given that it rained, what is the probability that the forecast was for rain?"
The problem arose because this question is the wrong way round. What you really want to know is, "Given that the forecast is for rain, what is the probability that it will actually rain?" The statistic here is much less impressive: about 33%.
Why did this happen?
