Arby's Promotion

In these bog posts, I try to come up with something that has struck my fancy in the last couple days. Regardless the fact that no one has used the phrase "struck my fancy" in a generation or two, You know what I mean.

A couple related things happened lately. Both were promotions by companies, both used probabilities correctly, but kind of sneakily, I guess hoping that people wouldn't notice. Both are good math applications for students that are just starting out in probability and can be seen in ways they are familiar with.

First one - I was in Arby's - a place I love. They had a sign trying to get customers to go on-line to fill out a customer satisfaction survey. If so,you could "RECIEVE 10 CHANCES TO WIN $1,000 DAILY". The hook is a person figures their chances of winning has gone up tenfold. It hasn't.

We don't know what are chances really are. Let's suppose they randomly chose one from eighty people that have gone on-line to take the survey. You have a one out of eighty chance of winning = 1.25%,

But wait, you have ten chances to win. So it was like you were standing in a line with seventy-nine other people. Now there are ten of you in line. But wait. There are ten of everyone else, too. Your chance of winning is not 1:80. It is 10:800. But wait. This is also 1.25%.

This technique seems to also be common in school raffles. You pay your money and they tear off three raffle tickets and hand them to you. Even if you know probability, you momentarily might think you're better off. You would be if you were the only one getting three tickets, but every entrant is getting three tickets as well. You're not getting cheated, you just aren't any better off. It's basically a marketing ploy. No harm done.

I've gone on long enough. I'll do my second example next week.

Cooking Turkeys

There is a book entitled 100 Essential Things You Didn't Know You Didn't Know About Math and the Arts, by John Barrow. It has some interesting things. Some of the 100 are a stretch as far a being connected to "the arts" is concerned". And sometimes the connection to mathematics is a little tenuous. But still, a good book.

It had one section dealing with instructions for cooking a turkey. It turns out that instructions are different in different parts of the world. We might be familiar with instructions such as this.

• Turkey weights of 8 to 11 pounds, roast for 2.5 to 3 hours.
• Turkey weights of 12 to 14 pounds, roast for 3 to 3.5 hours.
• Turkey weights of 15 to 20 pounds, roast for 3.5 to 4.5 hours.

These instructions are helpful, but there are some issues. A 14 pound turkey would get cooked for 3.5 hours, but so does a 15 pound turkey. So these instructions get you in the ballpark, but there is another way of expression these instructions. 

The British Turkey Information Service (Apparently a real thing). The directions are now in minutes and kilograms:

• Less than 4 kg in weight? Cook for 20 minutes per kilograms then add another 70 minutes at the end.
• More than 4 kg in weight? Cook for 20 minutes per kilograms then add another 90 minutes at the end.

The British system could be written like this:

• y = 20x+70, if x <  4
• y = 20x+90, if x  > 4

Mathematically speaking, we have gone from a piecewise defined function that resembles a step function to a piecewise function with y = mx+b parts. 

The British systems seems like it would give better cooking estimates. There are, however problems at x = 4. The function isn't even defined there. And if it was, should the cooking time be 150 minutes or 170, or split the difference and go for 160?

I would think some fine tuning could easily fix the problem. The BTIS needs to get on this. As it stands, looking at the functions and the graphs would make for interesting student activities.

Vietnam Memorial

The are many structures in the world that can be used to generate math applications. One of these is the Vietnam Memorial in Washington D.C. 

In 1980, individuals were invited to submit possible designs for the memorial. Over two thousands plans were submitted. The one selected was designed by Maya Ying Lin - at the time, an undergraduate architecture student at Yale University. She is currently an accomplished artist and designer. The wall was built in 1982. 

Its design is such that one wing points directly to the Washington Monument and the other to the Lincoln Memorial. At its greatest height, it stands 10 feet 3 inches. The top is at ground level, so the memorial itself actually sits below ground. Each wall is 246 feet 9 inches long. The walls meet at an angle of 125°12′. What look to be giant triangles are actually trapezoids with a long base of 10 feet three inches tapering to a small bases of 8 inches at each end.   

There are a number of math problems possible ranging from simple geometry to trigonometry. 

  

- How far is it from one end to the other? (Use the angle, sides of 246 feet, 9 inches and the law of cosines.)

- What is the area of the memorial? (Use the formula for area of a trapezoid.)

- What would be the length of one section of the memorial if extended to make a triangle? (This baffled me for a while. You can set up a proportion from the sides of the triangles that are formed to find a solution.)

Running vs Driving

My daughter, Karen, sent me this information about a different kind of running race. The race information is at http://www.wingsforlifeworldrun.com/us/en/santa-clarita-ca/.

You can go to the site, but here is the pertinent information -

1. Participants will begin running at 4 AM.
2. The catcher car will take off 30 minutes later, initially going just under 10 miles per hour.
3. Gradually, the catcher car will increase speeds throughout the race and pursue runners form behind.
4. Once the catcher car passes an individual, that marks their personal finish line.

This is a little like the infamous "A trains leave Omaha, with another train leaving an hour later..." type of problem. Those problems are kind of pointless, though. All you really care about is when your train gets in, and that is solved by just looking it up online. 

This one is more fun. If you are in the race, you really would want to do some math ahead of time to plan your strategy. The car gradually speeds up. Since we don't know how much, let's just assume it stays at 10 mph. If so, running at 10 mph or faster means we won't ever get caught. (There are runners that could go a long way at that pace. However, if the car just picks it up to 16 miles per hour, it is going at world record mile pace.)

Suppose you are running at a very respectable 8 miles per hour for "t" hours. The car is moving at 10 mph for "t-0.5" hours. Since distance = rate x time and we want to see when the car catches you (i.e., the distances are the same), we have the equation, 10(t-0.5) = 8t. Solving the equation gives t = 2.5 hours. Depending on how fast the car speeds up, you will be caught sometime within 2 hours, 30 minutes. 

It seems to me there are quite a few interesting problems the potential runner could come up with if he/she was going to compete in this race. Such as: When will you get caught if you go 7 mph; how fast do you have to run to not get caught for an hour; what happens if the car goes 2 mph faster every half hour? I'll leave that to you to consider.

Super Tuesday

Today happens to be Super Tuesday. A few days ago I saw a news release on cnn.com that I thought was interesting.

Cruz has the backing of 28% of Republican voters nationwide, unseating Trump, who won the support of 26% in the latest NBC News/Wall Street Journal poll. But Cruz's 2-point edge is within the poll's margin of error, and it's not clear if the survey captures real movement in the race or is simply an outlier. The results are a major change from last month's NBC News/Wall Street Journal poll, when Trump held a 13-point lead over Cruz, 33% to 20%.

"I have never done will in the Wall Street Journal Poll. I think somebody at Wall Street Journal doesn't like me, but I never do well in the Wall Street Journal poll," Trump said. "So I don't know. They do these small samples and I don't know exactly what it represents."

Another Wall Street journal poll released Wednesday found registered voters to be divided on whether the Senate should vote this year on a replacement for late Supreme Court Justice Antonin Scalia. Forty-three percent believe the Senate should vote on a replacement this year rather than wait for President Barak Obama's term to end, versus 42% who oppose a vote.

The NBC New/Wall Street Journal pollsters contacted 800 registered voters for the question on the Supreme Court, with a margin of error of  +/-3.5 percentage points, and 400 Republican primary voters for the Republican field, with a margin of error of 4.9 percentage points.

There are some good mathematical points to be made in this article.

• The first paragraph could lead to a discussion of what an outlier is. 
• The second paragraph shows that candidate Trump should probably do some research on how polls and "these small samples" work.
• The final paragraph speaks to the concept of margin of error. Note that the margin of error was greater when fewer people were polled.

The margin of error is usually not well explained when used in news reports and is generally misunderstood. A poll result of 43% with a margin of error of 3.5% would imply a range of 39.5 to 46.5%. However, the true result isn't necessarily within that range. Pollsters tend to state that the true result would be in that range with a probability of 90%, 95%, or 99%. The article doesn't say what that probability is.

We can figure it out, though. With a sample size of n, a 99% level of confidence yields a range of 1.29/sqrt n. A 95% level is 0.98/sqrt n. A 90% level of confidence is 0.821/sqrt n.

Most polls use a 95% level of confidence and that, in fact is the case here. Note that 0.98/sqrt(800) = 3.46% and that  0.98/sqrt(400) = 4.9%, which agree with the values given in the article.