A local politician claims that 1 in 5 automobile accidents involve a teenage driver. He is advocating increasing the age at which teenagers can drive alone. Over a 2 month period, there are 67 accidents in your city, and only 9 of them involve a teenage driver. If the politician is correct, what is the chance that you would have observed 9 or less accidents involving teenagers?

Question

A local politician claims that 1 in 5 automobile accidents involve a teenage driver. He is advocating increasing the age at which teenagers can drive alone. Over a 2 month period, there are 67 accidents in your city, and only 9 of them involve a teenage driver. If the politician is correct, what is the chance that you would have observed 9 or less accidents involving teenagers?

I have the correct answer, it's 0.0524, but I don't understand how to get this answer. Please help. (I would like to know how to do this by hand and also using a TI-84 calculator)

Thanks in advance.

Answer 1

To calculate the probability, we can use the binomial probability formula. The formula is:

P(X = x) = C(n, x) * p^x * (1 - p)^(n - x)

Where:
- P(X = x) is the probability of getting exactly x successes (in this case, the number of accidents involving teenage drivers)
- C(n, x) is the combination function, which calculates the number of ways to choose x items from a set of n items
- p is the probability of success for each trial (1 in 5 accidents involve a teenage driver)
- x is the number of successes
- n is the number of trials (total number of accidents in this case)

First, we need to calculate the probability of getting exactly 9 accidents involving teenage drivers out of 67 total accidents:

Using the formula:

P(X = 9) = C(67, 9) * (1/5)^9 * (4/5)^(67 - 9)

For the calculation, you can expand the combination function and use the appropriate calculator functions:

C(67, 9) = 67! / (9! * (67-9)!)

Alternatively, you can use a calculator or software that has a built-in function for calculating binomial probabilities.

For a TI-84 calculator, you can use the "binompdf" function as follows:

binompdf(n, p, x)

In this case:

n = 67 (number of trials)
p = 1/5 (probability of success)
x = 9 (number of successes)

Using the formula or the calculator function, you will get P(X = 9) = 0.0524100827, which can be approximated as 0.0524.

This means that the probability of observing 9 or fewer accidents involving teenage drivers, assuming the politician's claim is correct, is approximately 0.0524 or 5.24%.