Probabilistically zero events

In this short post, we show that events which are theoretically impossible to occur can be present. That is, practically there can be events which theoretically should not have occurred! Sounds fun?

Mathematically, if the probability that an event occurs is zero, does not mean that, that event cannot occur. Often, I cite the example of events that have continuous probability distribution. In such distributions, the probability that any point from sample space is selected is, in fact, zero. For example, consider [from wiki] a species of bacteria that have lifetime of about 4 to 6 hours. Thus, the probability distribution of the lifetime of these bacteria is continuous as it can take any value, say 2 hours, 7 hours, 3.0005 hours etc. Now, if we ask the following question - 

"What is the probability that a bacterium lives exactly 5 hours?".

This probability is, hold-your-breath, ZERO(0). Why? Because a lot of bacteria live for approximately 5 hours, but there is no chance that any given bacterium dies at exactly 5.0000000000... hours.

Instead, if we ask -

"What is the probability that a bacterium dies between 5 and 5.01 hours?"

This probability could be zero and can be found by integrating the probability density function of the lifetime from 5 to 5.01.

So, what did we see just now? We saw that even when an event that has probability zero (bacterium dying exactly at 5th hour), it could occur in practise (there would be bacteria that die at 5th hour)

And finally, the converse - Impossibility implies zero probability, is true!