Question

I have 5 umbrellas distributed between my house and my barn. Every morning I walk from my house to my barn and every afternoon I walk from my barn back to my house. I take an umbrella with me only if it is raining; if it is not raining I leave any umbrellas behind. It may happen that all the umbrellas are at one place, I am at the other, it is raining, I must leave, and so I get wet. The weather where I live is remarkably consistent and at any given point in time there is a 60/% chance of rain. To simplify things assume that the trip is a short one and I am lucky so that if I start a trip and it is not raining, then it never begins to rain while I am walking. I have been making this trip back and forth between my house and my barn every day for many years. On what percentage of those walks did I get wet? Here are some potential questions that you might have and their answers. Question: Do I ever take the umbrellas somewhere else? Answer: No, they are only stored in either the house or the barn and they only travel between the house and the barn as described above. Question: What was my first location, the location that I first walked from, the house or the barn? Answer: It doesn't matter. The problem is symmetrical with respect to the house and the barn. Question: What was the initial distribution of the umbrellas? In other words how many were in the house and how many were in the barn when I started making these trips all those years ago? Answer: It doesn't matter. The answer to question 1 will be the same no matter the initial distribution. This is also a hint as to what kind of Markov chain represents the physical situation. "I want to be certain to clarify something on problem 1 . I did not say it in the original statement of the problem, but if I am at one of the two locations, then I am never influenced about when to walk to the other by the weather. So, when it is time for me to leave, I leave whether it is raining or not and whether I have an umbrella or not."

Answer 1

To determine the percentage of walks you got wet, we use a probability approach. Given a 60% chance of rain, we calculate the probability of getting wet on each walk and sum them up. The total probability is 50%.

Step-by-step explanation:

To determine the percentage of walks where you got wet, we can use a probability approach. Let's consider the possible outcomes of each walk. There are two possibilities: either it is raining or it is not raining.

Given that there is a 60% chance of rain at any given point in time, the probability of it raining on a walk is 0.6. Therefore, the probability of it not raining on a walk is 0.4.

If we assume each walk is independent and there is an equal chance of starting at the house or the barn, we can calculate the probability of getting wet. There are four possible scenarios:

1. Start at the house, rain, no umbrella: This has a probability of 0.5 * 0.6 * 0.4 = 0.12
2. Start at the house, rain, umbrella: This has a probability of 0.5 * 0.6 * 0.6 = 0.18
3. Start at the barn, rain, no umbrella: This has a probability of 0.5 * 0.4 * 0.4 = 0.08
4. Start at the barn, rain, umbrella: This has a probability of 0.5 * 0.4 * 0.6 = 0.12

The total probability of getting wet is the sum of these four probabilities: 0.12 + 0.18 + 0.08 + 0.12 = 0.5.

Therefore, you would get wet on 50% of your walks.