Question

question 7 (1 point) saved
answer the question true or false.
in any experiment with exactly four sample points in the sample space, the probability of each sample point is .25.
true
false

question 8 (1 point)
solve the problem.
four hundred accidents that occurred on a saturday night were analyzed. the number of vehicles involved and whether alcohol played a role in the accident were recorded. the results are shown below:

number of vehicles involved
did alcohol play a role? | 1 | 2 | 3 or more | totals
yes | 57 | 96 | 17 | 170
no | 20 | 171 | 39 | 230
totals | 77 | 267 | 56 | 400

suppose that one of the 400 accidents is chosen at random. what is the probability that the accident involved alcohol or a single car?

Explanation:

Analyze Question 7

We are asked to evaluate the truth value of the statement: "In any experiment with exactly four sample points in the sample space, the probability of each sample point is .25."

A sample space \(S\) containing exactly four sample points can be written as \(S = \{e_1, e_2, e_3, e_4\}\). According to the fundamental axioms of probability, the sum of the probabilities of all sample points in a sample space must equal 1:

$$P(e_1) + P(e_2) + P(e_3) + P(e_4) = 1$$

However, there is no requirement that the sample points must be equally likely (i.e., symmetric). For example, we could have:

$$P(e_1) = 0.4, \quad P(e_2) = 0.3, \quad P(e_3) = 0.2, \quad P(e_4) = 0.1$$

Since the probabilities do not have to be equal, the statement is false.

Identify given values for Question 8

We are given a contingency table summarizing 400 accidents:

Did Alcohol Play a Role?	1 Vehicle	2 Vehicles	3 or more	Totals
No	20	171	39	230
Totals	77	267	56	400

We need to find the probability that a randomly chosen accident involved alcohol or a single car (1 vehicle).

Apply the addition rule for probability

Let \(A\) be the event that alcohol played a role.
Let \(B\) be the event that a single car (1 vehicle) was involved.

We want to find the union probability \(P(A \cup B)\). Using the addition rule:

$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$

From the table, we find the frequencies:

• Total number of accidents: \(N = 400\)
• Number of accidents involving alcohol, \(n(A) = 170\)
• Number of accidents involving a single car, \(n(B) = 77\)
• Number of accidents involving both alcohol and a single car, \(n(A \cap B) = 57\)

Calculate the final probability

Substitute the values into the probability formula:

$$P(A \cup B) = \frac{n(A) + n(B) - n(A \cap B)}{N}$$

$$P(A \cup B) = \frac{170 + 77 - 57}{400}$$

$$P(A \cup B) = \frac{190}{400} = 0.475$$

Answer:

Question 7

• True
• False (Correct answer)

Question 8

The probability that the accident involved alcohol or a single car is \(0.475\) (or \(\frac{19}{40}\)).