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This lecture defines zero-probability events and discusses some counterintuitive aspects of their apparently simple definition, in particular the fact that a zero-probability event is not an event that never happens.
The probability of an event has a value from 0 to 1 and is written as a fraction, a decimal or as a percentage. The notation for the probability of an event is P(event).
Let $A$ be an event, $\Pr$ be the probability measure. $A$ has zero probability if $\Pr(A) = 0$. $A$ is impossible if $A=\emptyset$. Impossibility implies zero probability, but the reverse is false. Consider the real line $\mathbb{R}$; if you randomly select a number $x$, the probability that $x=0$ is $0$, but this is not impossible.
An event having probability zero does not mean it is impossible, but that is unlikely to happen. If you look at the qoutients of all pairs of your numbers, and take the quotient of those which are equal by all pairs, for infitite many pairs this quotient will be zero.
The probability of an event is a number between 0 and 1 (inclusive). If the probability of an event is 0, then the event is impossible. On the other hand, an event with probability 1 is certain to occur.
A probability is a number that represents the likelihood of an uncertain event. Probabilities are always between 0 and 1, inclusive. The larger the probability, the more likely the event is to happen. A probability of 0 means that the event is impossible; it will never happen.
Rule 1: The probability of an impossible event is zero; the probability of a certain event is one. Therefore, for any event A, the range of possible probabilities is: 0 ≤ P (A) ≤ 1. Rule 2: For S the sample space of all possibilities, P (S) = 1.
