Question

In a city 20 percent of the population travels by car, 50 percent travels by bus and 10 percent travels by both car and bus. Then persons travelling neither by car nor by bus is (1) 80 percent (2) 40 percent (3) 60 percent (4) 70 percent

Short answer

40%

Step-by-step solution

- Define Variables

Let the total population be 100%. Let P(Car) be the percentage of people traveling by car, and P(Bus) be the percentage of people traveling by bus. Also, let P(Car and Bus) be the percentage of people traveling by both car and bus.

- Assign Given Values

From the problem, P(Car) = 20%, P(Bus) = 50%, and P(Car and Bus) = 10%.

- Use Inclusion-Exclusion Principle

The principle of inclusion-exclusion for two sets A and B states that P(A or B) = P(A) + P(B) - P(A and B). Here, P(A or B) represents the percentage of people traveling by either car or bus or both.

- Calculate Percentage of People Traveling by Car or Bus or Both

Using the given values: P(Car or Bus) = P(Car) + P(Bus) - P(Car and Bus) Substitute the given percentages: P(Car or Bus) = 20% + 50% - 10% = 60%.

- Calculate Percentage of People Traveling by Neither Car Nor Bus

To find the percentage of people traveling by neither car nor bus, subtract P(Car or Bus) from the total population. Therefore, P(Neither Car Nor Bus) = 100% - P(Car or Bus) P(Neither Car Nor Bus) = 100% - 60% = 40%.

Key concepts

probability principles

Probability principles help us to determine the likelihood of different outcomes. In this context, we are dealing with the probability of people choosing different modes of transportation. Probability can be expressed as a percentage or a fraction.
For example:

• The probability of traveling by car (P(Car)) is given as 20%.
• The probability of traveling by bus (P(Bus)) is 50%.
• The probability that someone travels by both car and bus (P(Car and Bus)) is 10%.

Understanding these basics is crucial for solving more complex probability problems. Always remember that at its core, probability is about understanding how likely an event is to happen.

inclusion-exclusion principle

The inclusion-exclusion principle is essential for solving problems involving overlapping sets or events. In this problem, we used it to determine the total percentage of people traveling by either car or bus or both.

According to the principle, for two events A and B:

• \[P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)\]

In simpler terms, you add the probability of each event happening individually and then subtract the probability that both events are happening at the same time.

Let's break down the formula based on our exercise:

• \[P(Car \text{ or } Bus) = P(Car) + P(Bus) - P(Car \text{ and } Bus)\]
• Substitute the given values: \[P(Car \text{ or } Bus) = 20\text{%} + 50\text{%} - 10\text{%} = 60\text{%}\]

Hence, 60% of the population travels by either car or bus or both. The inclusion-exclusion principle simplifies the process of calculating probabilities when there are overlaps between events.

percentage calculation

Percentage calculation is another fundamental math skill. In the context of probability problems, we often need to convert between percentages and fractions or vice versa.

Let's review the steps to calculate percentages in our problem:

• We started with the total population as 100%.
• We were given that 20% of the population travels by car, 50% by bus, and 10% by both car and bus.
• Using the inclusion-exclusion principle, we found that 60% travel by either car or bus or both.
• To find the percentage of people traveling by neither car nor bus, we subtracted this 60% from the total population: \[100\text{%} - 60\text{%} = 40\text{%}\]

Therefore, 40% of the population travels by neither car nor bus.
Remember, percentage calculations are often about understanding the part-whole relationship and tracking how the whole (in this case, the entire population) is divided among different parts (modes of transportation).