Question

PROBLEM SOLVING You take a bus from your neighborhood to your school. The express bus arrives at your neighborhood at a random time between 7:30 and 7:36 A.M. The local bus arrives at your neighborhood at a random time between 7:30 and 7:40 A.M. You arrive at the bus stop at 7:33 A.M. Find the probability that you missed both the express bus and the local bus.

Short answer

The probability that you missed both the express bus and the local bus is 0.15 or 15%.

Step-by-step solution

Determine the probability of missing the express bus

The express bus arrives within a 6 minute window (7:30 to 7:36). The current time is 7:33 AM now. That means there are 3 minutes left for the express bus to arrive. The chances of missing the express bus (based on it arriving in the remaining 3 minutes) would be \(\frac{3}{6} = 0.5\) or 50%.

Determine the probability of missing the local bus

The local bus arrives within a 10 minute window (7:30 to 7:40). The current time is 7:33 AM now. That means, there are 7 minutes left for the local bus to arrive. The chances of missing the local bus (based on it arriving in the remaining 7 minutes) would be \(\frac{3}{10} = 0.3\) or 30%.

Combine the probabilities

Since we want to know the probability of missing both the buses, we will multiply the individual probabilities together. Hence, the chance of missing both would be \(0.5 * 0.3 = 0.15\) or 15%.

Key concepts

Problem Solving

When you're solving a problem involving probability, especially in real-life situations like catching buses, it's important to start by understanding the details of the problem. In this scenario, you need to determine the likelihood of missing both an express and a local bus when you arrive at a particular time. This means focusing on understanding the timeframes and the randomness of the buses' arrivals. To do this effectively:

• Identify and break down each component, such as the time intervals and the presence of randomness.
• Assess the probability of each event independently.
• Combine these probabilities to find the overall likelihood of missing both buses.

Problem-solving in probability requires both logical thinking and sometimes a bit of mathematical calculation to provide a clear solution.

Express Bus

The express bus in our problem arrives at any random time between 7:30 A.M. and 7:36 A.M., creating a 6-minute window. When you reach the bus stop at 7:33 A.M., you're already in the middle of this time window. Here's how to think about it:

• The time from 7:33 A.M. to 7:36 A.M. is crucial since that is when you might still catch the express bus.
• If the bus arrives anytime before 7:33 A.M., you miss it.
• Hence, you need to figure out how many minutes after 7:33 A.M. it can arrive (in this case, 3 minutes). This is half of the total 6-minute window.

The probability of missing the express bus is then calculated by looking at this ratio of missed minutes to the total window of possible arrivals, resulting in a probability of 50% or 0.5.

Local Bus

The local bus offers a wider window of arrival from 7:30 A.M. to 7:40 A.M., giving a total of 10 minutes for it to appear. Arriving at 7:33 A.M. gives you more minutes to rely on catching this bus compared to the express bus:

• The remaining available time you would benefit from (7:33 A.M. to 7:40 A.M.) is 7 minutes.
• However, if the local bus arrived before 7:33 A.M., you'd have missed it again.
• This gives you a probability ratio of 3 missed minutes out of 10, or 0.3, representing a 30% chance of missing the local bus.

Understanding these wider windows often gives you better odds with local services versus stricter express schedules.

Time Intervals

Time intervals are essential in understanding how likely events are, given specific conditions. In probability problems like this, it's all about measuring chances across an interval. The key points here include:

• Recognizing that the bus times fall within defined intervals – 6 minutes for the express bus and 10 for the local bus.
• Your arrival at the stop means certain parts of these intervals are no longer options for catching the bus, affecting your probabilities.
• You assess how much of the interval you effectively missed by arriving late and use this portion to calculate your probabilities.

Accurate understanding of time intervals lets you better structure problems and solutions in probabilities involving arrivals, providing insights into outcomes and improved strategies.