Problem 27 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

What is the probability of arriving at a traffic light when it is red if the red signal is lit for \(30 \mathrm{sec}\), the yellow signal for \(5 \mathrm{sec}\), and the green signal for \(45 \mathrm{sec}\) ?

Step-by-Step Solution

Answer

The probability of arriving at a traffic light when it is red is \(\frac{3}{8}\).

Step 1: Find total duration of one complete cycle

To find the total duration of one cycle, we will sum up the duration of all three signals: red, yellow, and green. Let T be the total duration of the cycle: \[T = 30 sec + 5 sec + 45 sec\]

Step 2: Calculate the probability

Now that we have the total duration of one cycle (T), we can find the probability of arriving at the traffic light when it is red. We know that the red signal is lit for 30 seconds, so we can find the probability by dividing the duration of the red signal by the total duration, T: \[\text{Probability} = \frac{\text{Duration of red signal}}{\text{Total duration}}\]

Step 3: Plug in values and compute

Now we can plug in the values for the duration of the red light and the total duration of the cycle: \[\text{Probability} = \frac{30 sec}{30 sec + 5 sec + 45 sec}\]

Step 4: Simplify the expression

Next, we will simplify the expression by adding the durations: \[\text{Probability} = \frac{30 sec}{80 sec}\]

Step 5: Calculate the final probability

Now, we will simplify the fraction to obtain our final probability: \[\text{Probability} = \frac{3}{8}\] Therefore, the probability of arriving at a traffic light when it is red is \(\frac{3}{8}\).

Key Concepts

Probability Calculation

Probability is a fundamental concept in both mathematics and daily decision-making. When we talk about probability, we're essentially discussing the chance of a certain event happening. But how do we calculate it? It all starts with understanding the idea of 'outcomes' and 'total possibilities'.

In the context of traffic lights, every time you approach an intersection, there's an outcome for each color the traffic light could be. Calculating the probability involves comparing the duration of one outcome (like the red light being on) to the total time of all possible outcomes (the complete traffic light cycle).

To bring clarity to this process, breaking it down into easier steps is advisable. At its core, you identify the total duration, calculate the time for each outcome, and then divide the outcome duration by the total duration. When students follow these protocol-like steps, it simplifies what could be an overwhelming equation into manageable parts, illustrating the practical application of probability in a real-world scenario.

Traffic Light Cycles

Traffic lights operate in cycles if you haven't given it much thought before. If you've ever stopped at a red light, you're witnessing part of a cycle that ensures traffic moves smoothly and safely. Understanding these cycles isn't just for city planners or engineers; it's an applied math problem you can figure out.

The timing of each color in a cycle has been carefully calculated to balance safety and traffic flow. This isn't a guessing game but a mathematic process. For example, a common cycle includes time for red, yellow (or amber), and green. By adding up the duration that each light is on, we can comprehend the full cycle of the traffic light.

Good advice for students grappling with this concept: pay attention to the timing of traffic lights when you're out and about. Seeing the practical application firsthand can make the concept more relatable and less abstract, fortifying your understanding of how traffic light cycles are timed and managed.

Applied Mathematics

So many students wonder where in real life they'll use the math they're learning in class. Probability in traffic signals is a shining example of applied mathematics, which is the practice of applying mathematical methods to solve real-world problems.

In the problem at hand, we use applied math to determine the likelihood of a red light at an intersection. The mathematical model we create by analyzing the traffic light cycle is an abstraction of reality, but it's an incredibly useful tool. It can inform us how long we might wait at the light, or it could be used by city traffic engineers to modify signal timings to reduce congestion.

Encourage students to seek out these practical scenarios. Problems that might seem complex at first can often be made less intimidating and more tangible by looking for their applications in everyday life. For instance, noticing that probabilities can predict traffic patterns might just provide that 'aha' moment where abstract math concepts suddenly snap into focus with real relevance.