Select the correct answer.
Jacob is leaving his house to go to work and is running 5 minutes late. In the past, when Jacob has been 5 minutes late for work, he has taken an alternate route to work a total of 16 times. Jacob has also taken his normal route to work 27 times when running 5 minutes late. The table shows the outcomes for taking each route.
| | Alternate route | Normal route |
| :-------- | :-------------- | :----------- |
| Late | 8 | 9 |
| On time | 3 | 12 |
| Early | 5 | 6 |
| Total | 16 | 27 |
Based on the information in the table, if Jacob's goal is to be on time or early for work, should he take the alternate route or his normal route?
A. Jacob should take his normal route to work.
B. There is not enough information to determine if Jacob should take the alternate route or his normal route to work.
C. Neither route has an advantage.
D. Jacob should take the alternate route to work.
Answer (2)
Calculate the probability of being on time or early for the alternate route: 16 3 + 5 = 0.5 .
Calculate the probability of being on time or early for the normal route: 27 12 + 6 = 3 2 ≈ 0.667 .
Compare the probabilities: 0.5"> 0.667 > 0.5 .
Jacob should take his normal route to work: Jacob should take his normal route to work.
Explanation
Analyze the problem Let's analyze the problem. Jacob wants to maximize his chances of being on time or early for work. We have data on two routes: an alternate route and his normal route. We need to calculate the probability of Jacob being on time or early for each route and then compare those probabilities to determine which route is better.
Calculate probability for alternate route First, let's calculate the probability of Jacob being on time or early if he takes the alternate route. According to the table, he is on time 3 times and early 5 times out of a total of 16 times he takes the alternate route. So, the probability is calculated as follows: P ( " O n t im e or e a r l y "∣" A lt er na t e ro u t e " ) = Total number of times alternate route is taken Number of times on time + Number of times early P ( " O n t im e or e a r l y "∣" A lt er na t e ro u t e " ) = 16 3 + 5 = 16 8 = 0.5 So, the probability of Jacob being on time or early when taking the alternate route is 0.5 or 50%.
Calculate probability for normal route Next, let's calculate the probability of Jacob being on time or early if he takes his normal route. According to the table, he is on time 12 times and early 6 times out of a total of 27 times he takes the normal route. So, the probability is calculated as follows: P ( " O n t im e or e a r l y "∣" N or ma l ro u t e " ) = Total number of times normal route is taken Number of times on time + Number of times early P ( " O n t im e or e a r l y "∣" N or ma l ro u t e " ) = 27 12 + 6 = 27 18 = 3 2 ≈ 0.667 So, the probability of Jacob being on time or early when taking the normal route is approximately 0.667 or 66.7%.
Compare probabilities and conclude Now, let's compare the two probabilities. The probability of being on time or early using the alternate route is 0.5, while the probability of being on time or early using the normal route is approximately 0.667. Since 0.667 > 0.5, Jacob has a higher chance of being on time or early if he takes his normal route.
Examples
This type of probability calculation can be used in various real-life scenarios, such as deciding which route to take to work or school based on historical data, or determining which investment strategy is more likely to yield a positive return based on past performance. For example, if you are deciding between two routes to get to a meeting, you can track how often you are on time or early using each route. By calculating the probability of being on time or early for each route, you can make an informed decision about which route to take to maximize your chances of arriving on time.
Jacob should take his normal route to work, as it has a higher probability of getting him there on time or early (approximately 66.7%) compared to the alternate route (50%).
;
