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.

Prob(9 or less accidents involving teenagers)

= P(1 teenager involved) + P(2 teens involved) + ...+ P(9teens involved)
or
= 1 - P(no teen involved in 9 accidents)

Now when you say, "If the politician is correct" I will assume that we should use his/her data, and we would get.
Prob(teen involved) = .2
prob(no teen involved) = .8

P(as stated) = 1 - .8^9 = .86578

IF we use the actual data
P(teen involved) = 9/67 = .134328
P(no teen involved) = 58/67 = .865672 , (note they a add up to 1)

Prob(as stated) = 1 - (58/67)^9 = .72699

Somehow I think that I misunderstand the question

To solve this problem, you can use the binomial probability formula. The formula is P(X ≤ x) = ∑(x=0 to x) nCx * p^x * (1-p)^(n-x), where:

P(X ≤ x) is the probability of getting x or fewer accidents involving teenage drivers,
n is the total number of accidents (67 in this case),
x is the number of accidents involving teenage drivers (9 in this case),
nCx is the number of combinations of n things taken x at a time (nCx = n! / (x!(n-x)!)),
p is the probability of getting an accident involving a teenage driver (1/5 in this case),
(1-p) is the probability of not getting an accident involving a teenage driver.

To calculate this by hand:

1. Calculate nCx: nCx = 67! / (9!(67-9)!) = 67! / (9!58!)
2. Calculate p^x: (1/5)^9
3. Calculate (1-p)^(n-x): (1-(1/5))^(67-9)
4. Multiply the three results: nCx * (1/5)^9 * (4/5)^(67-9)
5. Repeat steps 1-4 for x = 0, 1, 2, ..., 9
6. Sum the results of step 4 for x = 0 to 9

To calculate this using a TI-84 calculator:

1. Press the MATH button, and then scroll down to PROB.
2. Select binomcdf( under PROB, and press ENTER.
3. Enter the values into the calculator as follows:
- First, the total number of accidents (67),
- Second, the probability of success (1/5),
- Third, the number of accidents involving teenage drivers (9).
4. Press ENTER.
5. The calculator will give you the cumulative probability P(X ≤ x), which in this case is the probability of observing 9 or fewer accidents involving teenage drivers.

By following these steps, you should be able to obtain the correct answer of 0.0524.

To answer this question, we can use the binomial probability formula, which calculates the probability of obtaining a specific number of successes in a fixed number of trials. In this case, the "success" is an accident involving a teenage driver, and the "failure" is an accident not involving a teenage driver.

Let's break down the problem and calculate the probability step-by-step.

Step 1: Define the variables.
- n: total number of accidents in the city over a 2-month period (67 in this case)
- p: probability of a teenage driver being involved in an accident (1 in 5 or 1/5, which is 0.2)
- x: number of accidents involving teenage drivers that we want to observe (9 in this case)

Step 2: Calculate the probability of observing exactly 9 accidents involving teenage drivers.
Using the binomial probability formula:
P(X = x) = (nCx) * (p^x) * (q^(n-x))
where "nCx" is the binomial coefficient and can be calculated as n! / (x!(n-x)!)

For our problem:
P(X = 9) = (67C9) * (0.2^9) * (0.8^(67-9))
First, calculate (67C9) = 67! / (9!(67-9)!).
Then plug in the values:
P(X = 9) = (67C9) * (0.2^9) * (0.8^58)
You can find the binomial coefficient and exponentiation using a calculator or by hand.

Step 3: Calculate the probability of observing 9 or fewer accidents involving teenage drivers.
To do this, we need to calculate the cumulative probability from 0 to 9:
P(X ≤ 9) = P(X = 0) + P(X = 1) + ... + P(X = 9)

When calculating this by hand, you can calculate each individual probability (P(X = 0), P(X = 1), ..., P(X = 9)) using the formula in Step 2, and then sum them up.

Using a TI-84 calculator, you can use the binompdf function to directly calculate P(X ≤ 9):
1 - binomcdf(n, p, x-1)
where n = 67, p = 0.2, and x = 10 for 9 or fewer accidents.

That's how you can solve this problem by hand and using a calculator.