Questions: For traffic that flows on a highway, the headway is the average time between vehicles. On four-lane highways, the probability P (as a decimal) that the headway is at least t seconds when there are 500 vehicles per hour traveling one way is given by P=0.80^t Under these circumstances, what is the probability that the headway is at least 15 seconds? (Round your answer to two decimal places.) P=

For traffic that flows on a highway, the headway is the average time between vehicles. On four-lane highways, the probability P (as a decimal) that the headway is at least t seconds when there are 500 vehicles per hour traveling one way is given by
P=0.80^t

Under these circumstances, what is the probability that the headway is at least 15 seconds?
(Round your answer to two decimal places.)
P=

Transcript text: For traffic that flows on a highway, the headway is the average time between vehicles. On four-lane highways, the probability $P$ (as a decimal) that the headway is at least $t$ seconds when there are 500 vehicles per hour traveling one way is given by \[ P=0.80^{t} \] Under these circumstances, what is the probability that the headway is at least 15 seconds? (Round your answer to two decimal places.) \[ P= \]

Solution

Solution Steps

To find the probability that the headway is at least 15 seconds, we need to substitute $ t = 15 $ into the given probability formula $ P = 0.80^t $. After substituting, we will compute the value and round it to two decimal places.

Step 1: Substitute the Value of $ t $

We are given the probability formula for the headway as $ P = 0.80^t $. To find the probability that the headway is at least 15 seconds, we substitute $ t = 15 $ into the formula:

\[ P = 0.80^{15} \]

Step 2: Calculate the Probability

Now, we compute the value of $ P $:

\[ P = 0.80^{15} \approx 0.03518437208883203 \]

Step 3: Round the Result

Next, we round the computed probability to two decimal places:

\[ P \approx 0.04 \]