Basic Probability Theory
James W. Lamb
Probability plays an essential role in finding out the truth, hypotheses being more or less
probable given the evidence.
Probability also plays a central role in making decisions,
the likelihood of the consequences of alternative courses of action being a key
consideration.
Thus, as Bishop Joseph Butler wrote in the 1700’s, “probability is the
very guide of life.”
Probability theory is the mathematical study of probability.
Its applications include
science (e.g. quantum mechanics and Mendel’s theory of genetics), statistics
(probability distributions), games of chance, decision-making (expected value),
investment analysis, actuarial forecasts, and blocking spam (Bayes spam filtering).
My
approach in this chapter is to solve a sequence of problems of increasing difficulty,
introducing the methods of probability theory as I go.
The final problem is not only really
hard, having tripped up many a Ph.D., but famous as well, making it into the New York
Times.
Problem One
: What is the probability of randomly picking the ace of hearts from a well-
shuffled standard deck of 52 cards?
Likewise, what are the probabilities of randomly
selecting the following: an ace, a heart, an ace or a heart, some card other than an ace
or a heart, a card that is either red or black, a card that is both red and black?
For those unfamiliar with the standard deck of cards, there are four suits (hearts,
diamonds, clubs and spades), each consisting of thirteen cards (ace, 2, 3, 4, 5, 6, 7, 8,
9, 10, Jack, Queen, King).
Hearts and diamonds are red in color; clubs and spades
black.
Calculating the probability of picking the ace of hearts is simple math.
With 52 cards,
each having the same probability of being selected (since the selection is random), the
probability of drawing the ace of hearts (or any other particular card) is 1/52.
The probability of drawing an ace is also easily computed.
With four aces in the deck
there are four chances of picking an ace out of 52 total possibilities.
So the probability
is 4/52.
The calculation implicitly uses what is known as the
Special Disjunction Rule
:
P(AvB) = P(A) + P(B), where A and B are mutually exclusive, i.e. there is no
possibility of both A and B being true.
Since drawing an ace means drawing a card that’s either the ace of hearts (AH), the
ace of diamonds (AD), the ace of clubs (AC) or the ace of spades (AS), the Special
Disjunction Rule yields:
P(AH v AD v AC v AS) = P(AH) + P(AD) + P(AC) + P(AS) = 1/52 + 1/52 + 1/52 +
1/52 = 4/52
Obviously, AH, AD, AC, and AS are mutually exclusive, since only one card is being
drawn.
The Special Disjunction Rule can also be used to calculate the probability of selecting a
heart: