Thursday, May 22, 2014

Boolean Algebra and Probability

Boolean Algebra is a field of algebra which formalizes the operations performed in logic, that is, on propositional statements (statements that are either true or false). This requires operations such as "AND", "OR" and "NOT" in order to compose complex statements from simpler ones, such as "it is raining" and "the road is wet" being composed into something like "NOT(it is raining) AND the road is wet". These operators work in the following way:

The statement "Math is awesome AND One Direction suck" is true because each of the simpler statements being joined together is true. If at least one of the simpler statements is false, then the whole thing becomes false.

The nice thing about this algebra is that it can be naturally extended from normal numeric algebra by representing a true statement with 1 and a false statement with 0. So then we can turn the statement "Math is awesome AND One Direction suck" into "1 AND 1" which is equal to "1". Why does it equal 1? Because for "X AND Y" to be true, both X and Y have be true individually. For example the statement "Math is awesome AND One Direction make great music" is false because one of those substatements is false. With the OR operator, as long as at least one of the substatements is true, the whole thing becomes true. For example "Math is awesome OR One Direction make great music" is true because one of the substatements is true. The NOT operator takes one substatement and inverts it, that is, if it was true then it becomes false and if it was false then it becomes true. For example "NOT Math is awesome" is false and "NOT One Direction make great music" is true. Numerically speaking, the following list summarizes these rules:

0 AND 0 = 0
0 AND 1 = 0
1 AND 0 = 0
1 AND 1 = 1

0 OR 0 = 0
0 OR 1 = 1
1 OR 0 = 1
1 OR 1 = 1

NOT 0 = 1
NOT 1 = 0

Can the be some normal arithmetic equations which will give us these outputs?

NOT is easy. What arithmetic equation returns 1 when given 0 and returns 0 when given 1?
NOT X = 1 - X
1 - 0 gives 1 and 1 - 1 gives 0.

AND is also easy. Just multiply the two operands together.
X AND Y = X × Y

OR is a little more tricky. This is because there is not single operation that gives the same outputs as OR. However we can easily derive an equation by using DeMorgan's laws which express an OR using AND and NOT only. The statement "in order to apply for the job you need a degree or five year's experience" can be expressed as "do not apply for the job if you do not have a degree and have less than five year's experience". Formally this is written as


which numerically becomes

X OR Y = 1 - ((1 - X)(1 - Y))
X OR Y = 1 - (1 - Y - X + XY)
X OR Y = 1 - 1 + Y + X - XY
X OR Y = Y + X - XY

This makes sense. The OR operator is the addition of the two substatements minus one in case they are both true. This will return 1 whenever at least one of the two substatements is a 1.

But it also makes sense on a higher scale. If you think about it, these rules are the exact same rules for probability. Probability is a generalization of Boolean algebra. Rather than just considering truth and falsity like Boolean algebra does, probability considers values between those two cases, such that 1 is certainty, 0 is impossibility and all the numbers in between are a degree of uncertainty, where the closer the number is to 1 the more certain the statement is.

In fact AND and NOT are defined in exactly the same way in probability. But isn't OR defined simply as X + Y? That is in fact only the case when X and Y are events which are independent, that is, cannot happen together. In this case X AND Y is always zero and hence can be dropped from OR's equation. The equation we have derived is useful for dependent events as well such as asking for the probability of "two fair dice resulting in an even number and a number greater than 3". In this case you can't just add the probability of a die being even (3/6) and the probability of a die being greater than 3 (3/6) as otherwise you get 3/6 + 3/6 = 1 which means that you are certain that this will happen. The problem is that it is possible for the two substatements to be both true, that is, a die being both even and greater than 3 (namely 4 and 6). To fix this just subtract the probability of this special case occurring (2/6) which gives 3/6 + 3/6 - 2/6 = 2/3.

Let's illustrate it with a Venn diagram:

The Venn diagram above is showing two events as a red and blue circle. They have some overlap between them which is coloured purple. If the red circle represented the chance of a die being even and the blue circle represented the chance of a die being greater than 3, then the purple overlap would be the numbers 4 and 6. Now we want to find the probability of the red OR the blue circle occurring. If we just add their areas together we end up with the purple overlap being added twice since it is common to both circles, but we just want to add it once. So we subtract it once after adding both circles, which will result in having just one purple overlap. This is what is happening when you use the equation X + Y - XY.