### 4. Probability Distributions: Probability Models

### Discrete Probability Models

Discrete Probability Model

If the sample space #\Omega# of a random experiment consists of a *finite* or *countable* set of outcomes, we can use a **discrete probability model **to assign probabilities to events.

A set is **countable **if we can label its elements with the numbers #1,2,3,\ldots# in some systematic way.

Discrete Model: Calculating the Probability of an Event

For a discrete probability model, the probability of an event is the *sum of the probabilities* of all the outcomes in that event.

Consider the random experiment of rolling two dice and counting the upward-facing dots.

Calculate the probability of rolling a total of #6#. Round your answer to #3# decimal places.

For this random experiment, there are #36# equally-likely outcomes:

\[\Omega = \left\{

\begin{array}{cccccc}

(1,1), & (1,2), & (1,3), & (1,4), & (1,5), & (1,6)\\

(2,1), & (2,2), & (2,3), & (2,4), & (2,5), & (2,6)\\

(3,1), & (3,2), & (3,3), & (3,4), & (3,5), & (3,6)\\

(4,1), & (4,2), & (4,3), & (4,4), & (4,5), & (4,6)\\

(5,1), & (5,2), & (5,3), & (5,4), & (5,5), & (5,6)\\

(6,1), & (6,2), & (6,3), & (6,4), & (6,5), & (6,6)\\

\end{array}

\right\}\]

Since all outcomes are equally likely to occur, each outcome has a probability of #\cfrac{1}{36}#.

For a *discrete probability model *such as this, the probability of an event is the *sum of the probabilities* of all the outcomes in that event.

\[\begin{array}{rcl}

\mathbb{P}(\text{Roll a total of }6) &=& \mathbb{P}\big[(1,5)\big]+\mathbb{P}\big[(2,4)\big]+\mathbb{P}\big[(3,3)\big]+\mathbb{P}\big[(4,2)\big]+\mathbb{P}\big[(5,1)\big]\\ &=&\cfrac{1}{36}+\cfrac{1}{36}+\cfrac{1}{36}+\cfrac{1}{36}+\cfrac{1}{36}\\&=&0.139

\end{array}\]