The three rules that we've studied are going to be enough to calculate the
probabilities for any combination of propositions in propositional logic,
because we've covered negation and then conjunction and then disjunction.
And if you use them in various combinations, you can cover all of
propositional logic, but it's still useful to look at one
particular case, namely the probability of something happening in a series.
Let's look at a simple example involving a very short series, only two events, and
contraception. Assume that you want to prevent
contraception, and that you use two different methods.
What's the probability of conception being prevented by using two methods at
once? And let's assume that the probability is
0.9 of the first method working alone, and the probability is 0.9 of the second
method working alone. How do you calculate the probability of
preventing conception when you use both methods?
Well, one move might be to say, let's add them
together, 0.9 + 0.9 is 1.8, so maybe the
probability of preventing conception using both is 1.8.
No way, of course not. The maximum probability is one.
Probabilities vary between zero and one, so 1.8 probability just doesn't make any
sense. That can't be right.
Another possibility is to say we're going to multiply the probability of preventing
conception by using both methods is 0.9 * 0.9 which is 0.81.
That can't be right either. Why not?
Because, that would mean that there's less chance of preventing conception by
using both methods, namely 0.81, than there was using one of the methods alone,
0.9. If used both methods, the probability of
preventing conception ought to go up. So how do we calculate the probability of
one of these contraceptive methods working and preventing conception when we
use them both? The right way to do this is to first of
all, look at the probability of one of these methods failing.
So instead of looking at the probability of it working being 0.9, we say the
probability of it not working is 1 - 0.9 or 0.1.
Now, what's the probability that they both fail?
Well, the probability that they both fail is then going to be 0.1 * 0.1 which is
0.01. And now, we can say what's the
probability of them not both failing? That's going to be 1 - 0.01 or 0.99.
So that's how we calculate the probability of preventing conception when
you use both methods. Let's state this method in a formal rule.
The probability that an event will occur at least once in a series of independent
trials is one minus the probability that it will not occur in that series.
Notice that we're assuming that the trials are independent.
This rule is only going to work when the trials are independent.
And, what you do, is you take the probability that it will not occur.
And subtract it from one, and that gives you the probability that the event will
occur at least once in the series or symbolically, it's one minus the
probability of not h raised to the n, the number of trials.
Because to get the probability that it will not occur in any of those trials,
you have to raise it to the power of the number of trials.
So in our case it was 0.1 * 0.1 = 0.01. But if you use three methods it would be
0.1 * 0.1 * 0.1, that means 0.1 raised to the third power.
Or if you use four methods, it would be 0.1 * 0.1 * 0.1 * 0.1, 0.1 raised to the
fourth power. So you can see how you can generalize
from our simple case of two contraceptive methods to use this formula to determine
the probability of an event occurring in a series of independent trials.
So, one more quick example. What's the probability of getting heads
on a coin, at least once if you flip it five times?
There is one, [SOUND] two. That didn't flip, I'm going to do that
again. There we go, three, [SOUND] four.
now I didn't show you whether it got heads or tails.
What's the probability that one of those was a heads?
Well, what's the probability for each flip that it was not a heads?.5.
There are five flips, raise it to the fifth power, 0.5 * 0.5 *
0.5 * 0.5 * 0.5. You can figure that out on a calculator.
I'm not even going to try to do it in my head right now.
And, then you subtract that from one and it gives you the probability that you get
at least one heads in five independent flips.
And notice what it does is if it's six flips, it's going to be even more likely,
and seven is going to be more likely, because you're going to be raising 0.5 to
the sixth power or the seventh power in order to apply the rule.
So, that's just another example of how you take this rule regarding series and
calculate the probability of the event occurring at least once in a series of
independent trials.