Addendum to the Previous Two Posts

Apparently the name of the game show where the math puzzle comes from is "Let's Make A Deal", and that even professors and statisticians get it wrong. The "statistics" required to solve the problem is extremely simple, but people have a lot of trouble defining what the problem is and not making faulty assumptions, such as the two choices are independent of each other, or that the two choices are even similar.

These kinds of mistakes happen all the time in everything including the path to Nobel Prize-winning medical research, which is part of the reason why it took twenty years to convince the medical community. One interesting part of the story is that they were unable to culture the bacteria, until accidentally one weekend the lab was too busy with other things to do what they had been doing with the plates, which was to throw them out after two days. After this discovery, they stopped throwing away the plates before the bacteria could grow. Ta-dah!

Not only is there the problem that medical people incorrectly conclude that inability to culture a bacteria is "proof" that it is not the cause, or doesn't exist, but apparently the efforts to culture bacteria and probably any other scientific procedure have to be questioned due to high chances of them just being done wrong without knowing.

Another thing was they were looking for corkscrew-shaped spirochetes, but the bacteria presented differently on the plates and biopsy specimens. Important things can be passed over because of assumptions.

It was also interesting that the two researchers described some elements of their background and skills which enabled them to accomplish Nobel Prize-winning work, including general training rather than specialized training, a tendency of thinking off onto what might be considered inconsequential tangents, and hobbies in engineering, photography, and drawing. The latter helped to see detail.

Math Puzzle: Do You Want To Change Your Mind?

My mom came across this math puzzle recently. I had already seen it and solved it as it was given as a homework assignment in high school senior year in our special math class. Nowadays, it would be difficult to give these kinds of problems as homework, with the Web and Google search.

You are on a game show. There are three doors. Behind one of the doors is the prize of a million dollars. There is nothing behind the other two doors. You get to choose one door to open. If the prize is behind that door, you win. After you choose a door but before it is opened, the game show host opens one of the other doors and shows that it is empty. He then gives you the option to change your mind. Do you go with your first choice, or change your mind?

In other words, what is the probability of winning if you stay with your first choice, and what is the probability of winning if you change your mind?

Any takers?

Math and Logic Puzzles: Truth-tellers and Liars

A well-known logic puzzle that I first encountered during elementary school in one of the puzzle books that my mom bought for us:

You are traveling on a road to a castle and come to a fork in the road where it splits into two paths. Standing at this fork in the road are two gatekeepers, one who always tells the truth, and one who always lies, but you don't know who is the truth-teller and who is the liar. You need to know which road leads to the castle. What question do you ask to find the road to the castle?

***

If that's too easy, this one is not from elementary school:

The Truth-teller, the Liar, and the Ambiguous

The page includes the answer, and an interesting discussion. Which I haven't bothered to actually understand in detail yet.

If only real life were as clear as math.

Math for Tailgaters

I had a bunch of stuff to mail, so I drove to the post office to drop them off, in my once or twice per month driving outing. When I thought to myself, the driver in front of me is going kind of slow, I thought that must mean I'm improving health-wise. Of course, I didn't do anything stupid, like the driver who ended up behind me a little while later.

I found myself behind a motorcyclist on the freeway, so I decided to leave about 50 feet of clearance. Not only because my sister's young friend was recently killed in a motorcycle accident, but because that is always the common sense thing to do. 50 feet to me is a rather normal spacing between cars on a freeway. For motorcyclists I leave more, because I ask what if he slips, because they do.

The driver behind me thought it would be cool to tell me that I'm doing some great wrongdoing to him, by first honking and then passing on the left and practically driving through the front of my car (at least that's how it looks from the driver's seat) right before he exited the freeway. And the point is...?

Sometimes it's tempting to make a bigger gap or drive slower when being honked at by a tailgater. But that's almost resorting to similar stupid behavior.

First of all, when going any speed anywhere, it doesn't matter if there's 5, 10, 20, or 100 feet of space in front of your car or the car in front of you. You're still going the same speed, and won't get anywhere any faster! (in any relevant units of time)

A Wily Blog recently posted how math is useful for pill poppers. Here is an even simpler math problem of everyday life:

If I were to have ridden on the motorcyclist's butt, leaving a mere 10 feet instead of 50 feet of space, and the cars on the freeway are moving at 50 mph, how much sooner would the tailgater on my butt get to his destination?

(50 - 10 ft) / (5280 ft/mi) / (50 mi/hr) * (3600 s/hr) = 0.54 s

It would save half a second. This doesn't change no matter how long the tailgater is riding on my butt, if other cars and especially cars in front of me are going the same speed. Unless you think you can drive through cars, which apparently he did try to do.

According to the three-second rule mentioned in drivers' ed, how much space should I leave between cars, when traveling at 50 mph?

(3 s) / (3600 s/hr) * (50 mi/hr) * (5280 ft/mi) = 220 ft

I would also argue that if drivers in general didn't tailgate as much, traffic would actually move faster, because driving too close causes more stop-and-go which slows everything down, if you notice the delay time domino effect between acceleration of a line of cars.