Who Wants to Be a Millionaire?

I thought about the Who Wants to Be a Millionaire money board as a good math application. I've thought how much pressure it would be to be a contestant. Once you get up to the big money, each step is a big deal. It's tempting to go on, even though you could lose everything.

The relationship clearly isn't linear. If it was, it would make the decision making a lot easier. But what is going on with these numbers? I thought it would be interesting to try to fit an equation to this set of numbers.

It might be good to first have students take a shot at it without any help. Most would probably see that it isn't linear. Most in fact would see a two and an exponent would be involved somehow. There could be a pretty even split, though, between 2^x and x^2.

It might help students to graph the values 0 (1,100), (2,200)...

Clearly, some kind of doubling is going on. It goes from 100 to 200. It takes a bit of a detour at 300, then 500, 1,000, 2,000, 4,000, 8,000, 16,000, 32,000, 64,000. Then a bump at 125,000, but then rights itself to the doubling pattern for the last few.

An astute student might think that the relationship is something like 2^x. However, when it doesn't double, it come up a little short of doubling. So, the base value might be a little less than 2.

By the way, you might have noticed that this is not in dollars. It must be the European version and those are pounds or Euros or something. I don't remember what that symbol means. I meant to do the dollar version. I didn't even know there was another. The dollar version has similar, but not identical numbers. This same project could be done with those numbers as well.

Anyway, most students are familiar with the game, so it might be a fun exercise for them. Next week I'll share some final results.

Car Rentals

This is a good, solid math application for a system of linear equations.

I rented a car a couple of weeks ago. I had to pay so much per day. It seems like it used to be that you had some kind of flat payment you had to pay as well. If that was the case, it doesn't seem to be any more.

Let's pretend it is the case, though. Otherwise, this example is not going to be all that great.

Car #1 - It rents for $50 a day with a flat payment of $70.

Car #2 - It rents for $42 a day with a flat payment of $120.

What is the better deal? The obvious answer is, "It depends". So we can do a little math to find out what he depends on. Two helpful equations will be:

y = 50x+70
y = 42x+120

There are many ways to solve this. Probably the easiest is the substitution method.

50x+70 = 42x+120
8x+70 = 120
8x = 50
x = 6.25 days

Students can discuss the concept of a break-even point. This situation doesn't actually have one as I doubt if any car rental company would keep track of quarter-days. When I was in school there was so much I didn't know. So this would be a good exercise in not just math, but how car rentals work and what thy might cost.

The graphing method would take a bit more work, but students would get a better idea of what was going on.

Pictured below is the darling I rented.

Scratch off

Here is another company promotion. A local car dealership sent out a flyer with scratch-offs. People can't resist scratch-offs, so that is a bit of a ploy in itself.

You can win:

• $25,000        Odds        1:25,000
• $500             Odds        1:25,000
• $2                 Odds        24,996:25,000     OR     $25,000        Odds        1:25,000
• $1,000          Odd          1:25,000

I know that third line looks a little funky, but that is how they had it in the ad. The connector between all of these are "or" so what is the point? I guess they're trying mess with your head again and make it look more likely that you will win something other than the $2 prize. (We'll give them a pass on the fact that these are actually probabilities and not odds.)

To the casual observer it looks like a pretty good chance of making a lot of money - either a 3 out of 4 or a 4 out of 5 chance. Of course, in actuality, the chance of getting anything good is 4 out of 25,000 = 0.016%.

So students could do a number of things to analyze this psychologically and mathematically. Another activity is to find the expected value:

0.00016 x $25,000 + 0.00016 x $500 + 0.00016 x $25,000 + 0.99984 x $2 +  0.00016 x $1,000
= $10.24

So, I guess there is no harm in trying for it. I did the scratching. It gave a code and then you had to call to see what you won. I'm sure this gives them another chance to talk you into buying a car - and to give you, undoubtedly, your $2.

My preference - If I do by a car, save the promotion and just take $10.24 off the price.