Ian's Fenton-inspired animation ideas

Five ideas for storyboarded cartoon strips; I may be well off the mark here! My ideas are for cartoon strips with a slight interactive element. I'll write "Scene 1" to describe the first box in a cartoon strip, etc.

[1] One on Bayes' Theorem. Bayes' Theorem is not really a theorem---what I am thinking of is questions like "given X what is the probability of Y".

Scene 1. There is a murder that is somehow covered up on screen, and six suspects appear. One of them definitely committed the murder, and the police consider each of the suspects to be equally likely to be the murderer!

scene 2. Some information for suspect 1 is supplied. For example, (following Fenton) both the murderer and suspect 1 have blood type XYZ. A third of the suspects have blood type XYZ.

(Meaning that suspect 1 committed the murder with probability 1/2.)

Scene 3,4,5,6,7. In each of these scenes, information is given on each of the suspects which allows you to assign a probability of being the murderer to each of them.

Scene 8. You now have the chance of selecting the person most likely to have murdered.

Scene 9. The murder is revealed and you discover whether you were right. (I am assuming here that the person most likely to be the murderer, was the murderer!)

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[2] One on the Monty Hall Problem. First, there can be an interactivity corresponding to the game show. Next, there can be a storyboard cartoon to explain why the probability of winning the money is 2/3 IF you switch boxes. I attach a crap looking interactivity with the sort of thing I have in mind.

I think that most of the confusion from this problem arises from stating it poorly. Here's a statement.

* There are 3 boxes, one of which conceals money. * You choose a box, and don't open it yet. Call it A. * I reveal one of the other two boxes to be empty. Call it B. I offer you the chance to switch your choice to the remaining box, C. * Do you switch boxes?

To convert this question to mathematics and work out probabilities correctly you must assume: (i) That the money is equally likely to be in each box. (ii) That in the third step, I DELIBERATELY REVEAL AN EMPTY BOX.

These statements should really be part of the problem. The first one might be assumed, but the second one causes sufficient confusion that it should it be stated explicitly. After all, if neither of us know in which box the money lies, there's no point swapping boxes. This point could be illustrated by having two interactivies:

[A] The usual Monty Hall set up.

[B] A variant in which, when the host opens a box, he RANDOMLY opens one of the remaining two boxes. One third of the time he will actually reveal the prize in this case.

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[3] Here's a story about a question I heard once and then I made up some variants on the problem. I really like it.

Scene 1. A man X is pictured meeting up with an old friend, another man A. Man A tells man X:

"I have two children. One of them is a girl. What is the probability that they are both girls?"

Scene 2. Man X is pictured looking a little puzzled after his first meeting. In this scene he meets another old friend, man B. Man B tells man X:

"I have two children. The older one is a girl. What is the probability that they are both girls?"

Scene 3. Man X looks more confused. Now he meets man C who tells him:

"I have two children. Here is a photo of one of them (shows photo of a girl). What is the probability that they are both girls?"

Scene 4. Man X looks very confused now. Meets man D who tells him:

"I have two children. One of them is a girl. She is called Ellie. What is the probability that they are both girls?"

Scene 5. Man X looks worn down. Meets man E who tells him:

"I have two children. One of them is a girl who has a beard. What is the probability that they are both girls?"

I like these problems. The answer to the first question is 1/3, rather than 1/2. The second one is 1/2. The third, fourth, and fifth questions depend on what you choose to deduce from the added information (they are more vague than the first two). For example, suppose you assume that nobody in the world gives both their children the same name. Then the fourth man has two children: Ellie and NotEllie. So all we need is the probability that NotEllie is a girl (which is 1/2).

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[4] Two envelope problem. This is a classic, but it didn't seem to be on Fenton's list. Here's the problem:

--- You are offered two envelopes, both containing a (positive rational) amount of money. One of the envelopes contains twice the amount that the other envelope contains. You choose one envelope and open it. Say it contains £N. Now you are offered the chance to switch to the other envelope. Do you take this chance? It seems pointless, however, the other envelope contains either 2N or N/2, with a 1/2 chance of each. Hence the expected value of the other envelope is (2N + n/2)/2 = 5N/4. So surely you should swap!

--- The answer to the question is, of course, that you shouldn't swap. The part that said there is a 1/2 chance of 2N in the other envelope is wrong. There is no basis for that assumption. I made up a similar question once which is easier to understand:

--- What is the probability that if you select a positive integer randomly it is odd? Is the answer 1/2? Well, if we write

1,2,3,4,5,6,...

it looks that way. However, if we list odd numbers two at a time, followed by a single even number, like so:

1,3, 2, 5,7, 4, 9,11, 6,....

then the answer looks more like 2/3.

So the answer is not 1/2. In fact, the question has no meaning, as you can't select a positive integer randomly.

As with all these problems, I am keen to point out that the confusion is NOT mathematical, it is more philosophical. I mean, discussion of a guy opening boxes is not maths, that's just surrounding garbage. Once the problem is reduced to maths, there is no ambiguity, and little confusion.

For example, if you choose a number randomly from {1,2,..,10}, mathematically all that means is that you assign the fraction 1/10 to each of the numbers 1,2,..,10. You can't do the same thing with all the positive integers, because there are infinitely many.

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[5] Arbitrage. Here's another idea that I like, that I have thought about before. It's about arbitrage, that is, the idea of making money for free. For example, if two men A and B have a race, and I find odds of 5:1 on A winning by one book-keeper, and I find odds of 10:1 on B winning by another book-keeper, then by betting on both A and B winning in the separate book-keepers, I am certain of making a profit.

Thus I suggest for a fifth animation we have a simple book-keeper interactivity. Various odds are shown on screen before some sort of match. The user has a bank account. They can choose to make bets. The idea is to look out for arbitrage opportunities. If they bet appropriately, then they can be sure of making money. Then another match occurs, and they have to bet again, and so forth (trying to make as much money as possible).