The probability of an event is the ratio of the number of outcomes associated with that event divided by the total number of possible outcomes:
$$ \begin{equation}\begin{aligned} P(A)=\frac{n(A)}{n(U)}\\ \end{aligned}\end{equation} $$Consider the case of flipping an unbiased coin. There are $2$ possible outcomes:
- landing a head or
- landing a tail
There is only one outcome associated with getting a tail thus the probability of getting a tail is: $$ \begin{equation}\begin{aligned} P(T)&=\frac{n(T)}{n(U)}\\ &=\frac{1}{2}\\ \end{aligned}\end{equation} $$
Consider the case of pulling a face card (King, Queen or Jester) from a deck of $52$ cards. There are a total of $12$ face cards which means that the probability of attaining a face card will be: $$ \begin{equation}\begin{aligned} P(F)&=\frac{n(F)}{n(U)}\\ &=\frac{12}{52}\\ &=\frac{3}{13}\\ \end{aligned}\end{equation} $$
The non-face cards are $40$ in total thus the probability of pulling a non-face card is: $$ \begin{equation}\begin{aligned} P(N)&=\frac{n(N)}{n(U)}\\ &=\frac{40}{52}\\ &=\frac{10}{13}\\ \end{aligned}\end{equation} $$
Improbability vs. Impossibility
An event is considered to be improbable if it has a probability of zero ($0$). It is important to note that an event being improbable means that it is highly unlikely to occur. This probability of $0$, however, does not mean that it is impossible for the event to occur.
An improbable event is not necessarily an impossible event.
Research Questions
Upon completing this section, you will be able to answer the following:
- What is the probability of an event?
- Can the probability of an event exceed $1$?
- Can an event have a negative probability?
- If an event is improbable (has a probability of $0$), does that mean that it is impossible for the event to occur?
- What is meant by conditional probability?