Imagine that a traffic intersection has a stop light that repeatedly cycles through the normal sequence of traffic signals (green light, yellow light, and red light). In each cycle the stop light is green for 27 s, yellow for 3 s, and red for 30 s. Assume that cars arrive at the intersection uniformly, which means that in any one interval of time, approximately the same number of cars arrive at the intersection at any other time interval of equal length. Determine the probability that a car arrives at the intersection while the stop light is yellow. Give your answer as a percentage precise to two decimal places.

Imagine that a traffic intersection has a stop light that repeatedly cycles through the normal sequence of traffic signals (green light, yellow light, and red light). In each cycle the stop light is green for 27 s, yellow for 3 s, and red for 30 s. Assume that cars arrive at the intersection uniformly, which means that in any one interval of time, approximately the same number of cars arrive at the intersection at any other time interval of equal length. Determine the probability that a car arrives at the intersection while the stop light is yellow. Give your answer as a percentage precise to two decimal places.

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Concept explainers

Continuous Probability Distributions

Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.

Normal Distribution

Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!

Question

**Traffic Light Probability Problem**

Imagine that a traffic intersection has a stop light that repeatedly cycles through the normal sequence of traffic signals: green light, yellow light, and red light. In each cycle, the stop light is green for 27 seconds, yellow for 3 seconds, and red for 30 seconds. Assume that cars arrive at the intersection uniformly, which means that in any one interval of time, approximately the same number of cars arrive at the intersection as during any other interval of equal length.

**Problem:**

Determine the probability that a car arrives at the intersection while the stop light is yellow. Give your answer as a percentage precise to two decimal places.

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**Solution Explanation:**

To find the probability, first calculate the total duration of one complete cycle of the traffic light:

- Green light duration: 27 seconds
- Yellow light duration: 3 seconds
- Red light duration: 30 seconds

Total cycle duration = 27 + 3 + 30 = **60 seconds**

The probability that a car arrives during the yellow light is given by the duration of the yellow light divided by the total cycle duration:

\[ \text{Probability (yellow)} = \frac{3}{60} = \frac{1}{20} = 0.05 \]

Therefore, the probability as a percentage is:

\[ 0.05 \times 100 = 5.00\% \]

Thus, the probability that a car arrives during the yellow light is **5.00%**.

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