Expected Value

The topic this week in Probability was Expected Value. Unlike many of the topics, which I have been exposed to before, this one was completely new to me.

It is such a useful idea! If McDonald's has 5 different Happy Meal toys (do they still do that? Or do unhealthy foods advertising to kids not allow it anymore?), and there is an equally likely chance you will get any one toy, how many should you expect to buy to get them all? The answer is about 12. Now, you may be lucky and get them in just 5 purchases, or you may get unlucky, and have it take many many more than that, but on average (say you and 1,000 of your closest friends are all collecting the toys) it should only take 12 purchases.

One of my favorite things to do at a store is buy a grab bag. If you know 20% of the grab bags are worth $100, and 30% are worth $50, and 50% are worth $10, is it a good idea to pay $25 to choose one at random? Expected value can help you inform your decision. On average, the return of a grab bag is $40, so paying $25 is a reasonable gamble.

Probability classes are generally a lot of coin flipping and dice rolling, so the real life application of expected value was what really made me interested in my topic. On a phone call with my Dad, who works in finance, I mentioned the topic we were covering, and he immediately started talking about how he uses expected value all the time at work. That made me smile, because students always ask about math "when am I going to use this?" It is nice to hear that someone really does!

What is the chance a probability instructor will email you back?

In my Probability class we have been working on a difficult conditional probability assignment for two (three?) weeks. The due date for it was extended because so many of us voiced that we were having a tough time with it. The assignment was meant to be a solitary assignment done over just one week, but the instructor felt that since this is an assignment used in the "brick and mortar" classes, we should also have the benefit of discussion. I was really glad for that, because I really struggled with one particular part of the problem.

Because there is just so much to get done trying to work full time and take two classes, I try to stay ahead of the due dates as much as possible. In this case, when the discussion board opened up, I found out that I was ahead of the class as far as my progress on the problem. My excitement over being able to discuss the problem was dampened because the discussion didn't really add much for me; I had worked all the beginning steps out last week, while others waited until this week to get started. (Let me add that I recognize that they are likely very busy too, and they have just chosen a different way to manage their time as far as when they do classwork. I know not everyone can front load their week like I do.)

So where was I to go for help? Email to the rescue! I am so pleased that our instructor decided to take the time to help me through the difficult steps, and I was able to turn the assignment in last night. We probably had at least 5 rounds of back and forth email over the past week. I am now well versed in writing out conditional probability notation. Solving the problem: no problem, just me and a trusty tree diagram, but the notation just killed me!

This isn't the first time an instructor has taken the time to respond to my email queries. In Algebra II I had a lengthy email chain with the instructor over an extension of a topic in the module, something beyond what the class covered.

So the chance you'll get an email back from my probability instructor? Experimental results say it is 100%