- Your quiz this Friday will be on probability.
- This reading concerns "conditional probability", which is the
probability of an event given that some other event has already
occurred. So it's sort of a "probability within a probability" problem,
and that just seems to make them twice as hard!
- We start with the plant problem:
Before going on vacation for a week, you ask your spacy friend to water your ailing plant. Without water, the plant has a 90 percent chance of dying. Even with proper watering, it has a 20 percent chance of dying. And the probability that your friend will forget to water it is 30 percent.
- What's the chance that your plant will survive the week?
- If it's dead when you return, what's the chance that your friend forgot to water it?
- If your friend forgot to water it, what's the chance that it'll be dead when you return?
- Let's start by considering the universe. We'll represent it in two
ways:
- Venn diagrams, and
- a Table.
- Now we need to think about this notion of conditional
probabilities. They're a kind of "side bet". You reduce your focus to
just a part of the universe, the part that has already come to
pass. Here's how we write them: for example,
The probability of D given W is written as
Now our author mentions "Bayes's Theorem", and, just so you know what that is, here it is. It relates the conditional probabilities to the probability that both of two different things happen in a set:
And because the order of the intersection (both happening)
The given probabilities are
What's the chance your plant will die? -
- Now our author suggests that, rather than go through all this convoluted stuff, we think in terms of "natural frequencies". Let's see if we get the same answer any easier. Let's assume that there are 100 plants in the experiment, all either dead or not-dead, watered or not-watered.