Intro to Induction

    Induction is the tool used by television sleuths the world over. It is the favored tool of Sherlock and real-life investigators, and induction is all about probability. 

    Induction is the discipline of logic that is dedicated to analyzing how strong an argument is. This is done by comparing probabilities of different hypotheses, compiling evidence, and formulating conclusions. Inductive arguments do not need to be correct, they just need to be probably correct.

The Method

    The way induction works is a little different from deduction. While deduction uses truth tables and proofs in order to prove that an argument is valid, induction uses probability to determine how strong the argument is. 

    Induction deals only with invalid arguments, but that doesn't make it less valuable or less meaningful. In fact, induction is where the scientific method comes from, so the 'whole' discipline of science is based on inductive reasoning. 

    The process used for analyzing inductive arguments is much simpler than what's used for analyzing deduction. However, coming up with the information necessary to analyze is much harder. For deduction, you simply need to analyze truth values. For induction, you need to come up with percentages. (Though admittedly this isn't always true.)

    The classic example is rolling a couple dice or a deck of cards. Basically, you ask yourself what the probability is rolling a 6. Well, on a six-sided die there are six sides (obviously), but only one of them is actually the number 6. So the probability of you rolling a six is (1/6). 

    If you roll two dice, you first need to ask yourself what the probability is rolling 2 sixes. If each individual die has a probability of (1/6), then the two dice together have a probability of (1/36). I'll walk you through the steps. 

    If each individual die has a (1/6) probability of rolling a six, but we are rolling two dice at the same time, then we can multiply the two probabilities together. Whenever you have two probabilities that are independent you can simply multiply the probabilities together. More on that in a couple paragraphs. 

    So, we are faced with a simple math equation. (1/6) x (1/6). When multiplying fractions, you multiply the numerator by the numerator, and the denominator by the denominator. In this case that gives us (1/36). Hopefully that all makes sense. Please ask questions if it doesn't. 

    

    Now, on having independent probabilities. Independant probabilities mean that the two probabilities do not interact at all. This is fairly easy to see when talking about dice.  No matter what your first roll is, it simply cannot influence the second roll. If you roll a three on the first die, you have every opportunity to still roll a three on the second. This holds true even if the two dice bump into each other. 

A Harder Example

    Let's move on to the deck of cards. The deck of cards can highlight dependent, independent, and mutually exclusive probabilities. We'll start simple.

    We'll start by asking a question. Here, that would look like "What is the probability of pulling out a king or a queen?" The difference being "king or queen" vs "roll two sixes." This is called mutually exclusive. This means that both could happen, but only one will. In induction, this is called a restricted disjunct. 

    A restricted disjunct simply means that you simply add the two probabilities together. There are 4 kings and 4 queens in a deck of 52. So, (4/52) + (4/52). This equals (8/52). This simplifies down to (4/26)→(2/13). 

    When adding fractions, you have to make sure that both denominators (the bottom number) are the same. So long as both denominators, you simply add the two numerators (top numbers) together and then put the denominator underneath it. 

    Now we'll move to an example of a general disjunct. This can be used at any time a disjunct is present. It's a little bit more complicated, so I'll put down the formula. The formula is P(1&2)=[P(1)+P(2)]-P(1 x 2).

    An example of that would look like this. "What is the probability of pulling out a jack or a red?" Hopefully, it's clear what is different here. The difference between this example and the last example is that it is very possible for you to pull out a red jack, but impossible to pull out a king queen. 

    This is an example of a probability that is not mutually exclusive. It looks a little bit more intimidating, but it's actually not. Here we simply plug the probabilities into the formula. That will look like the following:

P(jack & red) = [(4/52)+(26/52)]-[(4/52) x (26/52)]

P(jack & red) = (30/52)-[(4/52) x (26/52)]

P(jack & red) = (30/52)-(104/2,704)

P(jack & red) = (30/52)-(52/1,352)

P(jack & red) = (30/52)-(26/676)

P(jack & red) = (30/52)-(13/338)

P(jack & red) = (30/52)-(1/26)       (This is found by dividing 338 by 13)

P(jack & red) = (15/26)-(1/26)

P(jack & red) = (14/26)

P(jack & red) = (7/13)

    As you can see, it can be a bit of a process. I did show every little thing going on to illustrate the whole process so most of those steps can be condensed into one though. What this tells us though is that there is a (7/13) chance that we pull out a card that is either red or a jack. 

    The reason it's important to subtract out that joint probability (called an independent conjunction) is because if you don't then the probability of the red jack is double counted, skewing the results. 

    That was still pretty easy though, so let's try something even harder. Let's ask "What is the probability of pulling out both a red and a club?" We have now added a level of difficulty. We're no longer asking how likely we are to pull out just one card, but two now. Not only that, but this is what is called a dependant conjunction. Meaning that the two probabilities do interact. 

    There is a formula for this too. It goes; P(1&2)= P(1) x P(2|1). What (2|1) means in English is 'two assuming 1' which translates to [P(1) x P(2)]/P(1) in math. 

    So let's break it down. We'll solve (2|1) first. 

P(2|1)= [(26/52)x(13/51)]/(26/52)

P(2|1)= (338/2,652)/(26/52)

P(2|1)= (17,576/66,300)

P(2|1)= (8,788/33,150)

P(2|1)= (4,394/16,575)

P(2|1)= (338/1,275)

    To the keen observer, you will notice that mathematically you can cancel out the (26/52) in that equation. Leaving you with a very simple (13/51). I generally don't do this because technically there is a difference between (338/1,275) and (13/51). It's a very small difference, but sometimes it matters. 

Conclusion

    That's the basics of induction. In a later page I'll talk about Bayes Theorem. That's a way we can evaluate a hypothesis based on evidence. It's actually very cool, and not all that complicated. 

    

    Aside from that, there's not much more to the practice of induction. Using it in arguments can get a little complicated, but just like with deduction, after you get it down you can do almost anything with it. 

    As always, be sure to ask questions:)