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The three rules that we've studied are 
going to be enough to calculate the 

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probabilities for any combination of 
propositions in propositional logic, 

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because we've covered negation and then 
conjunction and then disjunction. 

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And if you use them in various 
combinations, you can cover all of 

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propositional logic, 
but it's still useful to look at one 

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particular case, namely the probability 
of something happening in a series. 

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Let's look at a simple example involving 
a very short series, only two events, and 

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contraception. 
Assume that you want to prevent 

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contraception, and that you use two 
different methods. 

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What's the probability of conception 
being prevented by using two methods at 

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once? 
And let's assume that the probability is 

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0.9 of the first method working alone, 
and the probability is 0.9 of the second 

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method working alone. 
How do you calculate the probability of 

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preventing conception when you use both 
methods? 

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Well, 
one move might be to say, let's add them 

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together, 
0.9 + 0.9 is 1.8, so maybe the 

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probability of preventing conception 
using both is 1.8. 

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No way, of course not. 
The maximum probability is one. 

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Probabilities vary between zero and one, 
so 1.8 probability just doesn't make any 

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sense. 
That can't be right. 

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Another possibility is to say we're going 
to multiply the probability of preventing 

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conception by using both methods is 0.9 * 
0.9 which is 0.81. 

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That can't be right either. 
Why not? 

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Because, that would mean that there's 
less chance of preventing conception by 

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using both methods, namely 0.81, than 
there was using one of the methods alone, 

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0.9. 
If used both methods, the probability of 

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preventing conception ought to go up. 
So how do we calculate the probability of 

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one of these contraceptive methods 
working and preventing conception when we 

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use them both? 
The right way to do this is to first of 

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all, look at the probability of one of 
these methods failing. 

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So instead of looking at the probability 
of it working being 0.9, we say the 

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probability of it not working is 1 - 0.9 
or 0.1. 

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Now, what's the probability that they 
both fail? 

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Well, the probability that they both fail 
is then going to be 0.1 * 0.1 which is 

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0.01. 
And now, we can say what's the 

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probability of them not both failing? 
That's going to be 1 - 0.01 or 0.99. 

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So that's how we calculate the 
probability of preventing conception when 

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you use both methods. 
Let's state this method in a formal rule. 

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The probability that an event will occur 
at least once in a series of independent 

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trials is one minus the probability that 
it will not occur in that series. 

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Notice that we're assuming that the 
trials are independent. 

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This rule is only going to work when the 
trials are independent. 

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And, what you do, is you take the 
probability that it will not occur. 

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And subtract it from one, and that gives 
you the probability that the event will 

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occur at least once in the series or 
symbolically, it's one minus the 

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probability of not h raised to the n, the 
number of trials. 

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Because to get the probability that it 
will not occur in any of those trials, 

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you have to raise it to the power of the 
number of trials. 

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So in our case it was 0.1 * 0.1 = 0.01. 
But if you use three methods it would be 

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0.1 * 0.1 * 0.1, 
that means 0.1 raised to the third power. 

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Or if you use four methods, it would be 
0.1 * 0.1 * 0.1 * 0.1, 0.1 raised to the 

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fourth power. 
So you can see how you can generalize 

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from our simple case of two contraceptive 
methods to use this formula to determine 

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the probability of an event occurring in 
a series of independent trials. 

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So, one more quick example. 
What's the probability of getting heads 

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on a coin, at least once if you flip it 
five times? 

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There is one, [SOUND] two. 
That didn't flip, I'm going to do that 

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again. There we go, 
three, [SOUND] four. 

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now I didn't show you whether it got 
heads or tails. 

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What's the probability that one of those 
was a heads? 

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Well, what's the probability for each 
flip that it was not a heads?.5. 

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There are five flips, 
raise it to the fifth power, 0.5 * 0.5 * 

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0.5 * 0.5 * 0.5. 
You can figure that out on a calculator. 

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I'm not even going to try to do it in my 
head right now. 

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And, then you subtract that from one and 
it gives you the probability that you get 

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at least one heads in five independent 
flips. 

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And notice what it does is if it's six 
flips, it's going to be even more likely, 

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and seven is going to be more likely, 
because you're going to be raising 0.5 to 

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the sixth power or the seventh power in 
order to apply the rule. 

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So, that's just another example of how 
you take this rule regarding series and 

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calculate the probability of the event 
occurring at least once in a series of 

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independent trials.