Not Probabilities

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%:

                                               

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:

                                                  

You can use this to easily solve problems like this:

“Not probability” question

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 not crashing on the way to school tomorrow, which is P(not E).  So we can rearrange the equation to get it in the form “P(not E) = something…”:

Certainties and impossibilities

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

                                              

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:

                                            

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.