As discussed above **discrete random variables **have a finite (or listable) range. What are their probability models? It's easy. In fact, you knew this before taking this course. Right! The **probability model of a discrete random variable** is its range and associated probabilities.

For example, in the first roll in the game of craps (roll a pair of dice) let *X = *the sum of the upfaces. Then the range of *X* is *2, 3, 4, ... ,12*. Now **ASSUME** that the dice are fair. Upon recalling the picture of the sample space, it is easy to determine the probability model of *X*. For example, the probability that *X = 3* means the probability that a *(1,2)* or a *(2,1)* comes up which is *(1/36) + (1/36) = 2/36*. Using the same reasoning for the other range items, we obtain the probability model for *X*:

Range 2 3 4 5 6 7 8 9 10 11 12 Probabilities 1/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36

Instead of a table, how about a picture of the probability model? Just plot the probabilities (vertical) versus the range (horizontal). Here is a crude plot:

. : . . : : : : : . . : : : : : : : : : . ---+---------+---------+---------+---------+---------+--- 2.0 4.0 6.0 8.0 10.0 12.0Alas, two more items. A little notation here is useful. For a discrete random variable

Note that, in general for any discrete random variable, *p(x)* is a fraction and the sum of all the *p(x)* (over the range of *X*) is 1.

Although the term probability model makes sense here, it is often not used in practice. Usually we call the probability model of *X*, the **distribution** of *X*. It is confusing since we used the term distribution with sampling distributions of Chapter 1. We will sort
this out later.

- 1.
- Let X denote the number spun on a fair spinner with the numbers 1, 2, and 3 on it. Determine the probability model of X.
- 2.
- In the last problem, suppose we spin the the spinner twice. Let S be the sum of the numbers spun. The range of S is 2, 3, 4, 5, 6. Use a tree diagram to determine the probability model of S.
- 3.
- Repeat the last problem if the spinner is spun 3 times.
- 4.
- Let X denote the number of aces in a 2 card hand drawn without replacement from a
*well shuffled*standard deck 0f 52 cards. Then the range of X is {0, 1, 2}. Use a tree diagram to determine the probability model of X. - 5.
- Repeat the last problem under sampling with replacement.
- 6.
- Let X denote the number of hearts in a 2 card hand drawn without replacement from a
*well shuffled*standard deck 0f 52 cards. Then the range of X is {0, 1, 2}. Use a tree diagram to determine the probability model of X. - 7.
- In the urn problem (with the balls well mixed ) discussed above, let Z denote the number of red balls in the sample (without replacement) of size 3. Use a tree diagram to determine the probability model of Z. What is the range of Z? Is it discrete or continuous?