# A Story and some probability

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.

1. #1 Jocelyn
January 8, 2009

You could consider power going out as three effectively instantaneous points of time per year, and look at the probability that any of the points happens while you’re on the tram. The first power outage or the second power outage or the third power outage could happen while on the tram, so you’d add the probability of you being on the tram for each one. This suggests a probability of power going out while on the tram at some point during the year of 3 * P(tram). The power outages are basically sampling your probability distribution of being on the tram.

I’m not sure how this fits with the multiplicative probability thing.

2. #2 meichenl
January 8, 2009

I think we need to be a bit more specific about “the probability of this occurring.” If your parents were to live forever, going down on the tram regularly and the power going out once in a while, the probability that they would eventually be on the tram when the power goes out is one – it would have to happen eventually.

A more precise question is, “What is the probability that, within the next year, someone will be on the tram when the power goes out?”

At any randomly-selected time, you’ve already calculated a 0.2% chance that someone will be on the tram. That means that for any given power outage, there’s a 0.2% chance that it will trap someone.

For three power outages in a year, this means there is a roughly 0.6% chance of getting stuck in a given year.

Notes: In addition to uncertainties in the individual probabilities (uncertainty in the percent of time spent on the tram and uncertainty in the number of power outages), the calculation also assumes the events are independent. The power may go out three times a year, but those three times are normally during lightning storms or blizzards or things. If your parents use the tram less during those times the probability of getting stuck is less than the straightforward multiplication implies. (see http://en.wikipedia.org/wiki/Statistical_independence)

Also, I cheated on the last part, where there are three events per year and I multiplied the 0.2% by three. You can’t just multiply the probability of getting trapped in any given power outage by the number of outages. That calculation actually yields the expectation value (http://en.wikipedia.org/wiki/Expectation_value).

Instead, you need 1 – probability of not getting trapped, or assuming the probability of getting trapped is 0.2% in any given power outage, the probability of not getting trapped in three power outages is 1 – 0.998^3.

However
(1-x)^n = 1 – n*x when x<<1,
so
0.998^3 = (1 – .002)^3 = 1 – .006 = 0.994
1 – 0.998^3 = 1 – 0.994 = .006
to well within the accuracy of the other parts of the calculation

3. #3 Matt
January 8, 2009

Joceyln’s comment about viewing power outages as effectively instantaneous events is a good one. What you want in this case is a Poisson random variable (look it up on Wikipedia for more detail) — it gives the probability of a certain number of events happening in an interval of time if a) you know the average rate of events and b) such events are independent. a) we can estimate — say, 10 outages a year (or equivalently as you put it, 10 outages per 5.26 x 10^5 minutes). So the rate is 10/(5.26 x 10^5) = roughly 2 x 10^-5 per minute. Taking your estimate of 6 minutes per round trip, that would mean that the rate is about 1.2 x 10^-4 per trip.

What you want to do now is to add up all the probabilities that correspond to 1, 2, 3, etc. power outages in a trip — this is a pretty approximate step and I’m a little sketched out by it but I think it works fine for unlikely events. Instead of doing an infinite sum let’s exploit the fact that all the probabilities for 0,1,2,… events sum to 1, and then the possibility of at least one power outage occurring in a trip is 1-(the probability that no power outages happen in a trip). I get a probability of 1.4 x 10^-5, or about 0.0014%.

That said, probabilities mean nothing when the event happens to you or a loved one. Glad to hear no one was hurt.

4. #4 tulcod
January 8, 2009

you forgot to mention that the tram could have influenced the power loss. maybe it sucks a lot of power and blew the power station? maybe it blew a fuse? the exact cause is needed to give a more precise probability, because this calculation is just nonsense.

5. #5 meichenl
January 8, 2009

Matt’s right that you want a Poisson distribution if you view the outages as independent random events. The calculation I gave would be in the case where there are definitely going to be three power outages this year, but they happen at random times. That’s goofy.

6. #6 Rhett
January 8, 2009

Thanks everyone for your input, I knew something didn’t seem right.

Rhett

7. #7 Uncle Al
January 9, 2009

All one requires to go downhill once is a brake. One might imagine that would be a continency in the design specs.