2.6 Probability of Complements

Last updated: March 8th, 20202020-03-08Project preview

Probability of Complements

So far we have learnt to calculate the probability of some events happening. Now we are going to think about the probability of an event not happening.

Remember the experiment of drawing a hand of $5$ poker cards and calculating the probability of getting a full house. We already calculated that $P(FH)\approx0.0014$.

Imagine that now we are interested in calculating the probability of not getting a full house.

Instead of calculating all the possible combinations of cards that are not a full house, we are going to take advantage of the fact that we know $P(FH)$.

Notice that getting a full house and not getting a full house are the only two options, since we either get a full house or we don't. Thus, they make up the whole sample space.

So we can use the formula that we learnt in the previous section: $P(FH)+P(notFH)=1$.

Working out from there, we get that $P(notFH)=1-P(FH)\approx1-0.0014=0.9986$.

Generalising: if we have an event $E$ and its complement $E'$, we can calculate $P(E')$ using the formula:


Of course this comes from the fact that since $E$ and $E'$ make the whole sample space, $P(E)+P(E')=1$. And from that same fact, we can conclude that:


One final remark: in the full house example we already knew the probability of getting the full house, so we used that to calculate the probability of the complement.

But in general, whenever we are facing the problem of finding the probability of an event $E$, it is always a good approach to ask ourselves if it is easier to calculate $P(E)$, or $P(E')$ and then using the formula.

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