To save on gasoline expenses, Edith and Mathew agreed to carpool together for traveling to and from work. Edith preferred to travel on I-20 highway as it was usually the fastest, taking 25 minutes in the absence of traffic delays. Mathew pointed out that traffic jams on the highway can lead to long delays making the trip 45 minutes. He preferred to travel along Shea Boulevard, which was longer (35 minutes), but rarely had traffic jams. Edith agreed that in case of traffic jams, Shea Boulevard was a reasonable alternative. Neither of them knows the state of the highway ahead of time.

After driving to work on the I-20 highway for 1 month (20 workdays), they found the highway to be jammed 3 times. Assuming that this month is a good representation of all months ahead, should Edith and Mathew continue to use the highway for traveling to work? How would you conclusion change for the winter months, if bad weather makes it likely for traffic jams on the highway to increase to 6 days per month?
How would your conclusion change if Mathew purchased a new smart-phone app that could show the status of the highway traffic prior to their drive each morning, thus reducing the probability of them getting into a jam down to only 1day per month (where on this day, the app showed no traffic jam, but a jam developed in the meantime as they were driving along the highway). Comment on how the conclusions of this problem will change if there was no uncertainty and the highway always had traffic jams, whereas Shea Blvd was always traffic jam free? Is this scenario realistic and why?

To determine whether Edith and Mathew should continue using the I-20 highway for traveling to work, we need to compare the average time it takes on the highway (including traffic jams) to the average time it takes on Shea Boulevard.

Given that Edith and Mathew drove to work for a month (20 workdays) and encountered highway traffic jams for 3 days, we can estimate the average time it takes on the highway:

Total time on the highway without traffic jams = 17 days * 25 minutes = 425 minutes
Total time on the highway with traffic jams = 3 days * 45 minutes = 135 minutes

Average time on the highway per day = (425 minutes + 135 minutes) / 20 days = 28 minutes.

On Shea Boulevard, the average time is 35 minutes. Comparing these, we can see that using the highway saves Edith and Mathew 7 minutes per day compared to Shea Boulevard.

Considering the situation in winter where traffic jams increase to 6 days per month, we need to recalculate the average time on the highway:

Total time on the highway without traffic jams = 14 days * 25 minutes = 350 minutes
Total time on the highway with traffic jams = 6 days * 45 minutes = 270 minutes

Average time on the highway per day = (350 minutes + 270 minutes) / 20 days = 31 minutes.

In this case, using Shea Boulevard would save 4 minutes per day compared to the highway.

Now, let's consider the scenario where Mathew has a new smart-phone app that reduces the probability of encountering a traffic jam on the highway to only 1 day per month. Since they drive for 20 days per month, this means that on 19 days there will be no traffic jam.

Total time on the highway without traffic jams = 19 days * 25 minutes = 475 minutes
Total time on the highway with traffic jam = 1 day * 45 minutes = 45 minutes

Average time on the highway per day = (475 minutes + 45 minutes) / 20 days = 26 minutes.

Using the app, the highway saves Edith and Mathew 9 minutes per day compared to Shea Boulevard.

In the scenario where the highway always has traffic jams and Shea Boulevard is always traffic jam-free, the conclusion is straightforward: Shea Boulevard would always be the better choice, without the need for any further calculations.

However, in reality, it is highly unlikely that the highway would always have traffic jams while Shea Boulevard is always traffic jam-free. Traffic conditions are dynamic and can vary based on several factors such as time of day, accidents, road maintenance, and other unforeseen events. Therefore, it is more realistic to assume that both the highway and Shea Boulevard would occasionally experience traffic jams, although their frequency may differ.

To determine whether Edith and Mathew should continue using the I-20 highway for traveling to work, we need to compare the expected travel time for both routes.

For the I-20 highway:
- Without traffic jams, the travel time is 25 minutes.
- With traffic jams, the travel time is 45 minutes.
- Considering that the highway was jammed 3 times in a month (20 workdays), the probability of encountering a traffic jam on any given day is 3/20 or 0.15.

For Shea Boulevard:
- The travel time is consistently 35 minutes.
- Assuming Shea Boulevard rarely has traffic jams, we can consider it as negligible.

To calculate the expected travel time for the I-20 highway, we multiply the travel time in each scenario by the respective probabilities:

Expected travel time for I-20 = (0.85 * 25) + (0.15 * 45) = 31 + 6.75 = 37.75 minutes

Comparing this with the constant travel time on Shea Boulevard (35 minutes), we can conclude that Edith and Mathew should continue using the I-20 highway as it has a lower expected travel time.

Now, let's consider the scenario during winter months where bad weather increases the traffic jams on the highway to 6 days per month. In this case, the probability of encountering a traffic jam on any given day becomes 6/20 or 0.3.

Expected travel time for I-20 during winter = (0.7 * 25) + (0.3 * 45) = 17.5 + 13.5 = 31 minutes

Comparing this with the constant travel time on Shea Boulevard (35 minutes), we conclude that they should still continue using the I-20 highway during winter months, as it still has a lower expected travel time.

Now, let's consider the situation where Mathew purchases a new smart-phone app that reduces the probability of encountering a jam down to only 1 day per month, which occurs as a jam develops while they're driving on the highway.

Expected travel time for I-20 with the smart-phone app = (19/20 * 25) + (1/20 * 45) = 23.75 + 2.25 = 26 minutes

Comparing this with the constant travel time on Shea Boulevard (35 minutes), it is even more evident that they should continue using the I-20 highway with the smart-phone app, as it significantly reduces the expected travel time.

Lastly, if we consider a scenario where the highway always has traffic jams, and Shea Boulevard is always traffic jam-free, the conclusion would be straightforward. In this case, Edith and Mathew should always choose Shea Boulevard as it provides a consistent travel time without any uncertainty of encountering traffic jams.

However, this scenario is not realistic because it is highly unlikely for a highway to always have traffic jams and for an alternate route to always be free of traffic jams. Traffic conditions are subject to various factors such as time of day, accidents, roadwork, and other unpredictable events. Therefore, it is more realistic to consider the probabilities and expected travel times in making decisions about which route to choose.