Final answer:
The question involves calculating the average time between cars passing on a highway using an exponential distribution model. For five cars per minute, the average time between cars is 12 seconds. The probabilities for specific intervals require the exponential distribution formula.
Step-by-step explanation:
The question is related to the topic of probability and exponential distribution in the context of real-world applications, specifically traffic flow on a highway. This involves calculating the average duration between cars and the probability of certain times between the passing of cars.
a. On average, how many seconds elapse between two successive cars?
If cars pass at an average rate of five cars per minute, then we can calculate the average time between cars by dividing the number of seconds in a minute by the average number of cars per minute:
60 seconds / 5 cars = 12 seconds per car.
b. After a car passes by, how long on average will it take for another seven cars to pass by?
Since there are 12 seconds on average between cars, for seven cars to pass, it would take 7 cars × 12 seconds/car = 84 seconds.
c. Find the probability that after a car passes by, the next car will pass within the next 20 seconds.
To find this probability, we would use the exponential distribution function, which is beyond the scope of this summary but can be calculated using a formula involving the average rate and the desired time interval.
d. Find the probability that after a car passes by, the next car will not pass for at least another 15 seconds.
Similarly, this probability can be found using the exponential distribution function for the complementary event that a car passes after 15 seconds.