I am going to tell a story and then calculate the probability of part of it happening. Really, it’s just an excuse for me to put this online (since no one got hurt).
My parents have this house on a lake, but it is really hilly. There is a dock down there, but it is like a bajillion stairs to get there. Good for exercise, bad for carrying a cooler full of beer (and ice). I don’t know where he found about it, but my Dad got these guys from Minnesota to install this tram thingy. Basically it’s a little train car with benches that rides down on two rails to the dock below. The thing is powered from a mounted electric motor at the top of the hill. It is not some huge thing.
Anyway, the lake had been REAL low on water and recently it had rained a lot. Their boat had already been stuck under the dock from the rise water the week before and it had just rained. So, my parents took the tram down to check on the boat. Half way down, the tram stopped working. It turns out that it was due to a power failure, but they didn’t know that. There they were, stuck. They decided to make their way down the hill (it is really steep and slippery). My Mom made it down, but my Dad was going down holding on the rails. He decided (don’t know why) to go in between the two rails. After slipping, he slid down the hill on his stomach until he became wedged underneath one of the cross bars connecting the two rails. After various attempts to free himself, they had to call the fire department. Good job fire department, job well done. Of that left my Mom to have to explain to the all the neighbors why the fire department was there.
Now on to probability. My father was thinking about how to prevent this in the future, but the question is: how likely is this to happen?
Essentially, two things have to happen at the same time. Someone has to be in the tram (going up or down) AND the power has to go out during that trip. If the power went out before or after the trip, you could just take the stairs and get some exercise (and not get stuck).
Disclaimer: Probability is not my strong suit. I often mess these things up. The probability of two events occurring at the same time is the product of their individual probabilities. So, the next step is the estimate the probability of 1) being on the tram and 2) having the power go out.
I am going to have to really just ball park this first one. I estimate that it takes about 3 minutes for the tram to make a trip down to the lake and 3 minutes back up. How often does someone use this tram? Well, in the summer it is used much more than in the winter. I am going to estimate an average of 3 trips (up and down) per week. This gives a total of about 156 round trips per year. The total time per year on the tram is 156*(6 min) = 936 minutes. What is the probability of being in the tram?
Note that there are about 5.26 x 105 minutes in a year.
Now for the probability of the power going out. I don’t need to know how long the power is out, just the probability of it going out. I am going to completely ball park that the power goes out 3 times a year. This maybe high, but oh well. I am not sure how to break this down. Maybe I should say 3 days out of the year so that the probability of the power going out is 3/365 = 0.008. So, the probability of these two things happening at the same time would be:
I don’t feel comfortable with this answer. I think I have calculated the probability that you will be on the tram on the same DAY the power goes out. Not the same time. Maybe if I gave a time interval for the power to go out. Suppose it takes 1 second for the power to go out. This would drastically change the probability of power failure to:
I am not too happy with this either. It seems I could arbitrarily choose any time interval and get all sorts of different probabilities. I told you I am not too good with probabilities. If this last probability is used, this means that the probability of getting stuck would be 1.68 x 10-10. That is pretty small. Remember, the whole point of this calculation was just to tell the story.
UPDATE: I was incorrect. See the comments below. Thank you commenters for your help.