What is a ‘not’ probability? It’s the chance of a
certain event *not* happening. Say there’s a 30% chance that it’s going
to rain tomorrow. What is the probability that it *won’t* rain tomorrow.
Common sense tells us that it is 70%.

Think about an event like it raining tomorrow.
It’s either going to rain tomorrow or it’s *not* going to rain tomorrow.
One or the other *has to happen*. This means that if we add the
probabilities of either one happening together, we should get 1, or 100%:

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You can apply this logic to any event you want. The probability of the event happening plus the probability of the event not happening adds together to give 1:

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You can use this to easily solve problems like this:

What is the probability of your bus not crashing on the way to school tomorrow if the probability of it crashing is 0.0001? |

Solution |

To solve this, we just need to use the ‘not’ equation: Now we need to decide what the event is. We have a probability of the bus crashing, so let’s make that the event: We’re interested in the probability of the bus |

### Certainties and impossibilities

A certainty is something that has a 100 % chance of
happening – it will happen *for sure*:

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If I have a bag containing 6 red marbles, and I pick one marble out, it is a certainty that I will pick a red marble.

Impossibilities are the opposite of certainties. They are events that have no chance whatsoever of happening – they will never, ever occur:

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If I have a bag containing 6 red marbles, then picking out a blue marble is an impossibility – there is no way it could happen.

### Impossibility statements

Think about the following statement:

Nothing is impossible.

I’ve heard this used as a motivational slogan to
get people inspired and to tell them to never give up. However, there is one
thing wrong with it. It says that there is absolutely nothing that can’t be
done, that *everything* is possible. However, if that is true, then the
following statement must be correct:

It is impossible to be impossible.

Uh-oh. We’ve just contradicted the first statement
by finding something that *is* impossible – the word itself! So the
statement “nothing is impossible” contradicts itself.