Final answer:
The probability that the time between vehicle arrivals is 10 seconds or less is approximately 63.21%, for 6 seconds or less it is approximately 45.12%, and for 32 or more seconds it is about 4.08%.
Step-by-step explanation:
The time between arrivals of vehicles at a particular intersection follows an exponential probability distribution with a mean of 10 seconds. The exponential distribution is defined as P(X < x) = 1 - e^(-λx), where λ (lambda) is the rate of occurrence, and P(X < x) is the probability that the time between arrivals is less than x seconds.
To find the probability that the arrival time between vehicles is 10 seconds or less, you would use the following steps:
- Calculate the rate (λ) from the given mean (mean = 1/λ), which is 1/10 seconds.
- Plug in the value of x (10 seconds) into the exponential distribution formula to calculate P(X <= 10).
- Calculate 1 - e^(-1) = 1 - 1/e = 1 - 0.3679 = 0.6321. After rounding, the probability is approximately 0.6321, or 63.21%.
Similarly, for the probability that the arrival time between vehicles is 6 seconds or less, plug in x = 6:
- e^(-6/10) = e^(-0.6).
- Calculate 1 - e^(-0.6) = 0.4512. The probability is approximately 0.4512, or 45.12%.
Lastly, for the probability of 32 or more seconds between vehicle arrivals, the calculation is P(X >= 32) = e^(-32/10) = e^(-3.2).
After calculation, P(X >= 32) ≈ 0.0408. Thus, the probability is approximately 0.0408, or 4.08%.