Question

the following table shows the distribution of murders by type of weapon for murder cases in a particular country over the past 12 years. complete parts (a) through (e)

(a) is the given table a probability model? why or why not?
a. no, the probability of all events in the table is not greater than or equal to 0 and less than or equal to 1
b. no, the sum of the probabilities of all outcomes does not equal 1
c. no, the probability of all events in the table is not greater than or equal to 0 and less than or equal to 1, and the sum of the probabilities of all outcomes does not equal 1
d. yes, the rules required for a probability model are both met.

(b) what is the probability that a randomly selected murder resulted from a rifle or shotgun?
p(rifle or shotgun) = (type a decimal rounded to three decimal places as needed)

Explanation:

Step1: Check if it's a probability model

For a table to be a probability model, two conditions must be met:

1. Each probability \(P(X)\) must satisfy \(0\leq P(X)\leq1\).
2. The sum of all probabilities \(\sum P(X)=1\).

Looking at the table, all probabilities \(0.478, 0.025, 0.003, 0.143, 0.132, 0.055, 0.134\) are between \(0\) and \(1\).
Now, calculate the sum:

$$
LATEXBLOCK0
$$

Since both conditions are met, it is a probability model.

Step2: Calculate the probability of rifle or shotgun

For two mutually - exclusive events \(A\) (rifle) and \(B\) (shotgun), the probability \(P(A\cup B)=P(A)+P(B)\) (addition rule for mutually - exclusive events).
Given \(P(\text{rifle}) = 0.025\) and \(P(\text{shotgun})=0.003\)

$$P(\text{rifle or shotgun})=0.025 + 0.003=0.028$$

Answer:

(a) D. Yes, the rules required for a probability model are both met.
(b) \(0.028\)