Probability: The basics with some examples of its application

I was inspired to create this thread after reading Chris L T521’s “Differential Equations Tutorial”.

While reading this thread, assumed is a basic knowledge of fractions, percentages and decimal numbers.

The basic idea of probability is applying a number to the chance of an event occurring. The number we use is always between and including zero and one. Zero representing no chance of an event occurring with one representing the event in question being certain to happen.

Let’s look at some notation to describe what was just discussed. We shorthand the “probability of event A occurring” as $\displaystyle P(A)$

Given the idea that a probability is always between zero and one then $\displaystyle 0\leq P(A) \leq 1$

Now we have talked about what a probability is and the restrictions associated let’s define how we find a probability.

Let’s say we have a universal set (the set of everything in question) of numbers or objects and call it $\displaystyle \epsilon$ . The probability of event $\displaystyle A$ occurring is defined as:

$\displaystyle P(A) = \frac{\text{The number of As in the set}}{\text{The total number of entries in the set}}=\frac{n(A)}{n(\epsilon )}$

Now let’s apply this.

Example 1: Consider an unbiased die with the numbers one to six.

a) Find the probability of rolling a 5

b) Find the probability of rolling a prime number

c) Find the probability of rolling an 8

We have $\displaystyle \epsilon = \{1,2,3,4,5,6\}$ and $\displaystyle n(\epsilon ) = 6$

a) $\displaystyle P(5) = \frac{\text{The number of 5s in the set}}{\text{The total number of entries in the set}}=\frac{n(5)}{n(\epsilon )} =\frac{1}{6} $

b) $\displaystyle P(prime) = \frac{\text{The number of prime numbers in the set}}{\text{The total number of entries in the set}}=\frac{n(prime)}{n(\epsilon )} =\frac{3}{6} $

c) There are no 8’s on the dice in question so

$\displaystyle P(8) = \frac{\text{The number of 8s in the set}}{\text{The total number of entries in the set}}=\frac{n(8)}{n(\epsilon )} =\frac{0}{6}=0 $

Example 2: Consider a deck of cards with no jokers.

a) Find the probability of drawing an ace

b) Find the probability of drawing a picture card

c) Find the probability of drawing a red card

d) Find the probability of drawing a black queen

As there are 52 cards in a deck so $\displaystyle n(\epsilon ) = 52$

a) $\displaystyle P(ace) = \frac{\text{The number of aces in the set}}{\text{The total number of entries in the set}}=\frac{n(aces)}{n(\epsilon )} =\frac{4}{52} $

b) $\displaystyle P(picture~ card) =\frac{n(picture ~cards)}{n(\epsilon )} =\frac{12}{52} $

c) $\displaystyle P(red~ card) =\frac{n(red~ cards)}{n(\epsilon )} =\frac{26}{52} $

d) There are four queens in the deck, one in each suit but we are only after the black ones.

$\displaystyle P(black~ queen) =\frac{n(black ~queens)}{n(\epsilon )} =\frac{2}{52} $

This concludes the first part of a series of posts on probability

Next I plan on looking at complementary probability, intersection and union of sets and conditional probability.