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.