He he he... Math is hard!

You call up a pet shop that you know has two puppies. You ask if at least one of the puppies is a male and they say "yes", what are the odds that the the other dog is also a male? Or rather that both dogs are male?

Interestingly I asked this question today and much to my surprise my sister got it right after a really short period of time. The answer is, oddly enough, one in three. Whereas almost everybody will say 50/50 and insist over and over again even if you just told other people who said that, that they were wrong.

There's a series of problems that fall into the same paradigm which are basically Monty Hall riddles, where known information is used to make a determination that change the odds from what we'd expect. And it turns out that *any* known information changes the odds. So when you picked Door Number 1 on the Monty Hall show and he opens Door Number 2, the odds that the prize is behind Door Number 1 is 1/3 and the odds that it's behind door number 3 is 2/3s. It's a strange math problem whereby the odds change because something is known about something else. Monty Hall wouldn't open the prize door. So we are given information that changes the odds.

In science, we start with a guess and then figure out what that guess would mean for the world and then check whether or not reality agrees with that guess. The most important step is that if reality is found to not agree that guess is wrong, it doesn't matter who suggested it or how interesting, beautiful, complex, simple or clever the guess is, it is wrong.

When we've checked a lot of things and find that we just can't prove this guess wrong over a very long period of time, it really does change the odds that that guess is correct, if we found a problem with the theory, it would be rejected. So when a scientific theory has withstood the test of time even if we are unaware of the particular tests it has withstood we can be more confident that it is true because being wrong would make it rejected. And if it's rejected, we can find a Monty Hall Door that hasn't been rejected yet which is almost certainly more likely to be correct, or with regard to scientific theory less likely to be wrong. Science is simply the least wrong guess we haven't managed to prove wrong yet. And we know it's the less likely to be wrong because we tried to prove it wrong and failed to do so thus far. This is why Einstein didn't dethrone Newton when his physics was accepted during the 20th century. Because up until then, Newton was the least wrong answer we had at the time. Now we have an even less wrong answer which can explain not only what Newton explains but more. To this end I highly recommend the Relativity of Wrong by Asimov.

The idea that knowing things about things change the odds falls a bit strangely into Hempel's paradox.

Hempel's raven paradox is basically stating that if we accept that we are given evidence to the statement "all raven are black" by the existence of a black raven, we should also accept that the existence of a green apple also provides evidence to the statement "all raven are black" because said statement is the same as saying "all non-black objects are non-raven objects" and a green apple confirms this given statement. The most accepted solution to the problem is to agree with the conclusion and suggest that the weight of evidence provided by green apples is very small because of the sheer range of non-black and non-raven objects. (As an amusing aside, there are albino ravens which are white). So really everything we know that fails to prove a statement wrong makes that statement a bit more likely to be true. Just as knowing that at least one of the puppies is male makes it slightly more likely that both puppies were male (after all we had a 1/4 chance of two male puppies before we confirmed that female/female wasn't an option). Failing to prove the theory male-male wrong by confirming female/female wasn't the case made the odds just a bit higher (though we seem to take the fact that at least one puppy is male to conclude one down, one to go and a 50/50 chance on the "other" puppy.)

The same is likely true with regard to evolution. A design is made more likely to be useful because an individual doesn't die when dying is possible. This is why the genotype (the DNA itself rather than what it codes for) which is insulated within the nucleus and doesn't interact with the outside world (there's transfer RNA and various other process that move this information around) can become very usefully complex and encode for very useful and functional proteins even though it has no access anything extra-nuclear. And this is also likely how our minds work by being able to make guesses about the world and by failing to be surprised by the wrongness of those guesses manage to build an internal view of the outside world without needing the eyes to take and process information (which is the model we are usually given). It simply isn't necessary to process information coming in from the eyes, there's only the need to see if that agrees with what we think it should be and react if we're wrong. You can build models of the outside world with guesses and checking if those guesses mesh with reality. That's the essence of science, evolution, and how our minds work. The part that understands, learns, encodes, works, theorizes, and discovers need not be actually connected to reality to understand it. It can simply trudge along failing to be wrong, and as a consequence become more reliable.

As a crazy person on the Daily Show once pointed out, either the LHC blows up the world or it doesn't. So the odds are 50/50. It's logic like this that makes me wonder how I could make money off such things.

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If I roll two 6-sided dice and say that at least one of them is a 1. What are the odds that they are both 1s?

1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 3 3 4 4 4 4 4 4 5 5 5 5 5 5 6 6 6 6 6 6
1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6

Because we know one is a 1, the odds that the other is a 1 is different than the typical 1/6 that people would tend to believe. Because we don't know which one was the 1 and there's only one possibility of snake-eyes (1-1).

1 1 1 1 1 1 2 3 4 5 6
1 2 3 4 5 6 1 1 1 1 1

These are the only possibilities and they are all evenly likely to occur. There's 11 of them and only in 1 of them are they both 1s. So knowing that at least one die is a 1 makes the odds that "the other one" (we don't know what other is because we don't know which is the principle one, and we should rather say "both of them") a 1 in 11 chance. We'd figure the odds to be 1 in 6, but really they are far longer than that and nearly double. This makes for a pretty good idea in a gambling game. I'll get some dice roll them somewhere, look and tell the other person what "at least one of them is X" and have them make wager that pays 8 fold that the other one is the same. They should think they are getting an advantage paying a 8X on a 1/6 chance when really they are getting paid 8X on a 1/11. I'll be rich!

Update:

Strangely due to the setup there the odds are actually 1/6. Since I would have to give information about what one die is, the chance that the other die is the same as X, is 1/6 or the odds that you roll doubles. Whereas you have twice as many things to have the dice be if you roll non-doubles. So the game would have to be setup, where you ask "Is there a 1?" and then go from there. There are 10 rolls for which the answer would be yes, and only 1 where the answer would be yes and the other one is a 1 too. Whereas if I simply offer what one die is (even if you don't know which die, it makes the odds of the other die 1/6 to be anything including X).

If I tell you that there is one 1. The odds that the other is 1 is 1/6. If you ask if there's a 1, and the answer is yes, then the odds are 1/11. The odds that I'll have a die to tell you the value of is 100%. The odds that you guess a value is lower when I've rolled doubles, so when you get that question right, the odds are 1/11th. Just as if I told you as a petshop owner that I have at least one male puppy thereby negating the chance that I have female/female puppies. As if I had female/female puppies, I would have volunteered that I had a female puppy rather than a male.

Go figure, math is hard.