Probability Distribution

What is a Probability Distribution Anyway?

Imagine you're playing a game of rolling dice. A probability distribution is like a complete list of all possible outcomes when you roll the dice, plus it tells you how big the chance (probability) is for each outcome to appear. Simply put, it's a list of possibilities and how often they happen.

Getting to Know the Sample Space (All Possible Outcomes)

The sample space is like the collection of all outcomes that can happen in an experiment. We usually write it using the symbol $\Omega$ (Omega).

For example, if you roll a standard die with $6$ sides: The numbers that can show up are $1, 2, 3, 4, 5,$ or $6$ . Well, all these possibilities gathered together are called the sample space.

\Omega = \{1, 2, 3, 4, 5, 6\}

There are a total of $6$ possible outcomes here.

Events and Their Probabilities

What is an Event?

An event is one or more specific outcomes that we are interested in from the sample space. An event is a smaller part of the sample space.

Example:

From rolling one die earlier ( $\Omega = \{1, 2, 3, 4, 5, 6\}$ ), let's say we want to look at the event "getting an even number". The even numbers there are $2, 4,$ and $6$ . So, the event of getting an even number (let's call it event A) is:

A = \{2, 4, 6\}

There are $3$ outcomes in this event A.

Calculating the Probability of an Event

Probability is a number that shows how likely an event is to happen. Calculating it is easy:

P(\text{Event}) = \frac{\text{Number of outcomes in that event}}{\text{Total number of outcomes in the sample space}}

Using the symbols from our example:

P(A) = \frac{|A|}{|\Omega|}

Where:

$|A|$ means the number of outcomes in event A (there were $3$ , right?).
$|\Omega|$ means the total number of outcomes in the sample space (there were $6$ , right?).

So, the probability of the event "getting an even number" (event A) is:

P(A) = \frac{3}{6} = \frac{1}{2}

This means the chance is half-and-half, or $50\%$ .

Rules of the Game for Probability Distributions

Probability distributions have two important rules:

The probability of each outcome ( $P(x)$ ) must be a value between $0$ and $1$ . It can't be negative or greater than $1$ .

$0 \leq P(x) \leq 1$
If you add up all the probabilities for every outcome in the sample space, the total must be exactly $1$ .

$\sum_{x \in \Omega} P(x) = 1$

Rolling One Die

If we roll one fair die (meaning each side has an equal chance of showing up), the distribution looks like this:

Outcome $x$	Probability $P(x)$
$1$	$1/6$
$2$	$1/6$
$3$	$1/6$
$4$	$1/6$
$5$	$1/6$
$6$	$1/6$

Why are they all $1/6$ ?

Because there are $6$ sides, and the die is fair, so each side has $1$ chance out of the total $6$ possibilities. Try adding up all the probabilities: $1/6 + 1/6 + 1/6 + 1/6 + 1/6 + 1/6 = 6/6 = 1$ . It fits the second rule perfectly!

Rolling Two Dice

Now imagine rolling two dice, say one red die and one white die. If we list all the possible pairs of numbers that can appear, there will be $36$ possibilities!

Why $36$ ?

Because the first die has $6$ possibilities, the second die also has $6$ , so the total is $6 \times 6 = 36$ pairs.

The resulting pairs can be written like this:

Red Die 1, White Die 1 -> (1, 1)
Red Die 1, White Die 2 -> (1, 2)
... and so on until ...
Red Die 6, White Die 6 -> (6, 6)

Each of these pairs has an equally small probability, which is $1/36$ .

Important! Distinguish between (1, 2) and (2, 1)!

If the dice are different colors (like red and white), getting a $3$ on the red die and a $2$ on the white die is different from getting a $2$ on the red die and a $3$ on the white die. So, the order matters if the dice can be distinguished.