All the different kinds of probability are governed by the same basic rules.
A priori probabilities are governed by those statistical probabilities follow
these rules, subject to probabilities should follow these rules.
But what are these basic rules? We're going to look at four basic rules.
The first one governs simply negation. So you toss a coin and you get tails.
What's the probability that you get tails?
We said one in two or.5. Now,
what's the probability that you do not get tails?
Well, that means that's the probability that you get heads.
So it's also 5.. It's one minus the 5. probability of
tails gives you the.5 probability of not getting tails, that is, of getting heads.
Well, that's really simple, because there are only two possibilities.
What about a die? You roll a die and you get a two.
What's the probability that you'd get a two?
Well it's one in six. What's the probability that you would not
get a two? Well, not getting a two is getting either
one, or three, or four, or five, or six, so there are five possibilities of not
getting a two, and that means that, that five times out of six you do not get a
two. So the probability of not getting a two is one minus the probability of
getting a two. The probability of getting a two is one
in six, so the probability of not getting a two is five in six.
So our rule is basically that the probability that an event will not occur
is one minus the probability that the event will occur.
And we can put that in symbols like this. You put PR for probability and then in
parentheses you put what it's the probability of.
We're going to use H for hypothesis or the event that we're testing the
probability for. And the little squiggle or tilde means
not, just like it did when we were talking
about propositional calculus a few weeks ago.
So, the probability of not the hypothesis, is equal to one minus the
probability of the hypothesis. That's basically the rule for
probabilities of negations. And we explained why it holds, because
certainty is one. The thing happening and the thing not
happening have to add up to one, because it's gotta either happen or not happen,
so one minus the probability that it happens is gotta be the probability that
it does not happen. So hope now the rule makes sense, and
that you can apply it in a couple of exercises.