Question

A traffic count at a highway junction revealed that out of 5,000 cars that passed through the junction in 1 week, 3,000 turned to the right. Find the probability that a car will turn (a) to the right, and to the left.

Short answer

The probability of a car turning right at the highway junction is \(\frac{3}{5}\), and the probability of a car turning left is \(\frac{2}{5}\).

Step-by-step solution

Understand the given information

We are given the total number of cars that passed through the junction (5,000) and the number of cars that turned right (3,000).

Calculate the probability of turning right

To find the probability of a car turning right, we will divide the number of cars that turned right (3,000) by the total number of cars that passed through the junction (5,000): \(P(\text{turning right}) = \frac{\text{number of cars that turned right}}{\text{total number of cars}} = \frac{3,000}{5,000}\)

Simplify the fraction

We can simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor (1,000): \(P(\text{turning right}) = \frac{3,000}{5,000} = \frac{3}{5}\)

Calculate the probability of turning left

Since there are two possible directions for a car to turn (right or left), the probability of a car turning left is equal to 1 minus the probability of a car turning right: \(P(\text{turning left}) = 1 - P(\text{turning right}) = 1 - \frac{3}{5}\)

Simplify the fraction

To simplify the fraction, we can first find a common denominator (5): \(P(\text{turning left}) = \frac{5}{5} - \frac{3}{5}\) Now, subtract the numerators: \(P(\text{turning left}) = \frac{5-3}{5} = \frac{2}{5}\)

Final Answer

The probability of a car turning right at the highway junction is \(\frac{3}{5}\), and the probability of a car turning left is \(\frac{2}{5}\).

Key concepts

Traffic Analysis

Traffic analysis is a technique used to collect, assess, and understand data about traffic flow at a specific location, like a highway junction. In our exercise, the analysis was based on cars passing through a junction over a week. Key insights include the total number of cars (5,000) and how many turned right (3,000). Understanding such data helps us predict traffic patterns and make informed decisions. For example, towns might use it to improve road safety and efficiency.

In this scenario, we’re interested in determining the likelihood or probability of cars turning in specific directions. From the traffic count data, calculating probabilities becomes straightforward by applying basic math principles. This lays groundwork for planning traffic control measures and optimizing junction design.

Fraction Simplification

Fraction simplification is essential in probability to make the results more understandable. In our exercise, we start with the fraction \( \frac{3,000}{5,000} \) representing the cars turning right. Although it is technically correct, simplifying is preferable.

We simplify fractions by dividing the numerator and denominator by their greatest common divisor (GCD). In this case, the GCD is 1,000. Applying this gives us:

• \( \frac{3,000 \div 1,000}{5,000 \div 1,000} = \frac{3}{5} \)

Now, this fraction \( \frac{3}{5} \) is much easier to interpret.

Fraction simplification makes mathematical expressions cleaner and more accessible, an important skill in all math-related tasks.

Step-by-Step Solutions

Step-by-step solutions are a fantastic tool to break down complex problems into manageable parts. In our traffic probability problem, the solution began by clearly understanding the problem—knowing how many cars passed and turned which direction.

The solution continued by calculating the probability that a car turns right, initially resulting in the fraction \( \frac{3,000}{5,000} \). By simplifying this to \( \frac{3}{5} \), it became easier to handle.

Once the probability of turning right was calculated, finding the probability of turning left was straightforward using complements. Since cars can either turn right or left:

• \( P(\text{turning left}) = 1 - P(\text{turning right}) = 1 - \frac{3}{5} \)

After simplifying, we arrived at \( \frac{2}{5} \). Ensuring each step is carefully executed is vital, and it reinforces understanding of probability concepts.