Probability Is Fun
Do you know what one of my favorite things in the world is? No you don’t, let me tell you. Probability. Once upon a time, I was a math major, and probability was my favorite class. I even tutored this Japanese guy in probability. Come to think of it, I think he failed. Oh well, can’t win ‘em all. For a while, I considered being an actuary. These are the people who spend all day analyzing risk for insurance companies. It would have been hours and hours of endless fun, but I figured I would get more chicks as an engineer. My current job involves a lot of thinking. It’s hard to describe, but there are lots of algorithms to develop, test, and have meetings about. I love my job, but sometimes I worry that all the thinking will make my head explode, so I take short breaks every now and then. What do I do to refresh on these short breaks? Sample problems from the actuarial exam! I am not making this up.
You can learn how to become an actuary at the be an actuary website, and you can find the sample test here and solutions here. Let’s do a sample problem together. It will be fun, I promise. Here is a question from the test:
A tour operator has a bus that can accommodate 20 tourists. The operator knows that tourists may not show up, so he sells 21 tickets. The probability that an individual tourist will not show up is 0.02, independent of all other tourists. Each ticket costs $50, and is non-refundable if a tourist fails to show up. If a tourist shows up and a seat is not available, the tour operator has to pay $100 (ticket cost + $50 penalty) to the tourist. What is the expected revenue of the tour operator?
Break into small groups and discuss this problem. Be prepared to present your solution at the end of this blog post. If you know anything about probability, this is a pretty easy problem to solve. If not, it isn’t so easy. That’s the great thing about probability. It seems hard, but is actually not hard. This is useful for looking smarter than you are.
Let’s solve the problem. I know that the expected value of any situation is the sum of each value multiplied by the probability of that value occurring. First I would like to know the probability of all 21 tourist showing up. This is easy, it’s 0.9821, which is about 0.65. This is because you can find the probability of multiple independent events occurring by multiplying the probability of those events together. So there is a 98% chance of any random tourist showing up, and a 65% chance of all 21 tourist who bought tickets showing up. This first time I did this, I calculated the probability as 0.0221, which would be the probability of all tourist not showing up. It took me like 2 minutes to figure out what I had done wrong. I’m so dumb.
Now we need to know the probability of 20 or less tourist showing up. We could calculate the probability of every combination of less than 20 tourist showing up and then sum all the probabilities, but that is a lot of work. We know that the sum of the probabilities of all possibilities is 1, and if 21 tourist don’t show up, then 20 or less must have shown up. So the probability of 20 or less tourist showing up is 1-.65, which is .35. So there is a 35% chance that 20 or less tourist will show up.
If 21 tourist show up, the tour operator takes in $950 (21 x $50 is $1050 minus a $100 penalty is $950). If 20 or less tourist show up, the tour operator will take in $1050, because nobody gets a refund. So the expected revenue of the tour operator is 0.65 x $950 + 0.35 x $1050, which is $985. So on average, the tour operator will take in $985 per tour. Do you see a problem here? You should. If the tour operator only sold 20 tickets, he would never have to pay a penalty, and would take in $1000 per tour, which is an average of $15 more per tour. Not only that, he wouldn’t have to ruin some poor tourist’s vacation. What a dumb ass.
Now for a couple editorial comments. There are no such thing as independent events. When somebody says, “assuming independent events”, 90% of the time it is a bad assumption. Think about the problem we just discussed. What are the odds that just one tourist won’t show up? Not likely, because most people don’t go on vacation by themselves, so it is significantly more likely that a group of tourist won’t show. Not only that, if all the tourist show up, one tourist is going to have to miss the tour while the rest of his or her group goes on the tour. I can’t stress this enough, tour operator = jerk. The assumption of independence only works for trivial problems, and real life problems are rarely trivial. Unfortunately, you often have to pretend that events are independent, because doing otherwise is too hard and beyond the scope of this blog.
Another complaint, this whole problem assumes that the tour always sells out. (It must be awesome!) This assumption should really be stated in the problem. Do you hear me actuarial exam problem making people? Also, I assume that this is another bad assumption. As fun as probability is, it is often one bad assumption after another. Probability = life.
Okay, if you are still reading, you must leave a comment. There is something seriously wrong with you, and I will find you help.
The image in this post is Einstein Was Wrong by _mpd_ and is licensed under Creative Commons.
