Unit 13 Discrete Probability Distributions

Elementary Statistics

Triola, Elementary Statistics 11/e

Unit 13 Discrete Probability Distributions

A random variable is a variable that has a single numerical value, determined by chance, for each outcome of a procedure.

Consider rolling dice. The sum of the dice is a number between 2 and 12. Until you roll the dice, you do not know which number is going to come up. This number is therefore a random variable. The event is the roll of the dice, and the random variable is a number between 2 and 12.

A probability distribution is a two-column table. The first column lists the values of the random variable. In the dice example, this would be a number between 2 and 12. The second column lists the probability of getting the value of the random variable in an event. For example, there are two ways to roll a three in dice out of a possible 36 combinations. Therefore, the probability of rolling a three is 2/36. That probability would be associated with the random variable 2. Here’s the whole table.

X / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / 10 / 11 / 12
P(X) / / / / / / / / / / /

When you roll the dice, one of these rolls has to occur. Hence the probability of getting a 2 or a 3 or … a 12 is

Knowing this, what do you suppose all the cells in the second row add up to?

Question #1.
What do the probabilities in a probability distribution table add up to?

Question #2.
What is the probability that we roll a 5?

Question #3.
What is the probability that we roll a 5 or a 6?

Question #4.
Histogram the probability distribution above. Use relative frequencies. Print it out and hand it in with your name on it.

Notice that this graph is symmetrical, it starts low, rises to a peak and then descends, but it is nota normal distribution. Now take a look at the bar over the bin equal to 7.

Question #5.
What is the height of the bar?

Question #6.
If we assumed that the width of the bar was equal to 1, what would be the area of the bar?

Question #7.
Now, recall from previous work, what is the probability of throwing a 7?

Hmm, that’s interesting. Compare the answers in questions #6 and #7.

Question #8.
Now figure out what is the probability of throwing a 5 or a 6 or a seven.

Question #9.
What is the sum of the areas of the bars over the bins corresponding to 5, 6 and 7?

What can you now conclude about a relationship between probability and area under the histogram?

This is the end of Unit 9. In class, you will get more practice with these concepts by working exercises in MyMathLab.