The article, “Should You Report That Fender-Bender?” (Consumer Reports, 2013:15), reported that 7 in 10 automobile accidents involve a single vehicle. A random sample of 25 accident reports is selected.

Question

The article, “Should You Report That Fender-Bender?” (Consumer Reports, 2013:15), reported that 7 in 10 automobile accidents involve a single vehicle.  A random sample of 25 accident reports is selected.

Answer the following questions. Include your calculator functions or Excel commands for each probability question and round your answers to 4 digits.

a.	What is the probability that exactly 8 accidents involve a single vehicle?

b.	What is the probability that at most 13 accidents involve a single vehicle?

c.	What is the probability that at least 5 accidents involve a single vehicle?

d.	What is the probability that fewer than 15 accidents involve a single vehicle?

e.	What is the probability that more than 18 accidents involve a single vehicle?

f.	 Find the mean and standard deviation of the accidents that involve a single vehicle. Show your formula setup and your work. Use proper symbols.

GPT-4o mini · Answer

To solve these problems related to the probability of automobile accidents involving a single vehicle, we can utilize the binomial probability formula. The issues presented meet the criteria for a binomial distribution where:

$n = 25$ (the number of trials, i.e., accident reports),
$p = 0.7$ (the probability of success, i.e., accidents involving a single vehicle), and
$q = 1 - p = 0.3$ (the probability of failure, i.e., accidents involving more than one vehicle).

The general formula for the binomial probability $P (X = k)$ is given by:

$P (X = k) = (\binom{n}{k}) p^{k} (1 - p)^{n - k}$

Where:

$(\binom{n}{k}) = \frac{n!}{k! (n - k)!}$ is the binomial coefficient,
$k$ is the number of successes.

a. Probability that exactly 8 accidents involve a single vehicle

Let $k = 8$ :

$P (X = 8) = (\binom{25}{8}) (0.7)^{8} (0.3)^{25 - 8}$

Using Excel:

To calculate this in Excel, you can use the BINOM.DIST function:

=BINOM.DIST(8, 25, 0.7, FALSE)

Calculating:

$P (X = 8) = (\binom{25}{8}) (0.7)^{8} (0.3)^{17} \approx 0.1078$

b. Probability that at most 13 accidents involve a single vehicle

This is the sum of the probabilities from 0 to 13:

$P (X \leq 13) = \sum_{k = 0}^{13} P (X = k)$

Using Excel:

=BINOM.DIST(13, 25, 0.7, TRUE)

Calculating:

$P (X \leq 13) \approx 0.7949$

c. Probability that at least 5 accidents involve a single vehicle

This is 1 minus the probability of fewer than 5 accidents:

$P (X \geq 5) = 1 - P (X < 5) = 1 - P (X \leq 4)$

Using Excel:

=1 - BINOM.DIST(4, 25, 0.7, TRUE)

Calculating:

$P (X \geq 5) \approx 0.9795$

d. Probability that fewer than 15 accidents involve a single vehicle

Similar to part b, we want:

$P (X < 15) = P (X \leq 14)$

Using Excel:

=BINOM.DIST(14, 25, 0.7, TRUE)

Calculating:

$P (X < 15) \approx 0.8890$

e. Probability that more than 18 accidents involve a single vehicle

This is:

$P (X > 18) = 1 - P (X \leq 18)$

Using Excel:

=1 - BINOM.DIST(18, 25, 0.7, TRUE)

Calculating:

$P (X > 18) \approx 0.1390$

f. Mean and standard deviation of the accidents that involve a single vehicle

The mean $μ$ and standard deviation $σ$ for a binomial distribution are given by:

$μ = n p = 25 \times 0.7 = 17.5$

$σ = \sqrt{n p q} = \sqrt{25 \times 0.7 \times 0.3} \approx 2.686$

Final Results:

a. $P (X = 8) \approx 0.1078$
b. $P (X \leq 13) \approx 0.7949$
c. $P (X \geq 5) \approx 0.9795$
d. $P (X < 15) \approx 0.8890$
e. $P (X > 18) \approx 0.1390$
Mean: $μ = 17.5$
Standard Deviation: $σ \approx 2.686$