Chapter 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 #3#. 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.
\[\begin{array}{rcl}
\mathbb{P}(\text{Roll }3) &=&\mathbb{P}\big[(1,2)\big]+\mathbb{P}\big[(2,1)\big]\\\\&=&\cfrac{1}{36}+\cfrac{1}{36}\\\\&=&0.056
\end{array}\]
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