Probability Review

Sample Space

A sample space is a set consisting of all possible outcomes of a random experiment. A sample space can be either discrete or continuous. When the sample space has a finite number of possibly outcomes, as in a coin flip, it is called a discrete sample space. The sample space for a coin flip consists of two outcomes when you ignore the unlikely possibility of a normal coin landing on its edge:

S = {0,1} = {H,T}
When the sample space has an infinite number of possible outcomes, that is, the number of outcomes equals the number of points on a given interval of the x axis, it is called a continuous sample space.

The distributive law of probability algebra can be applied to the computation of a sample space containing mutually exclusive outcomes. That is, when outcome A and outcome B are mutually exclusive, the sum of the probabilities for obtaining each of them separately equals the probability for obtaining either outcome A or outcome B, as shown by

P(A OR B) = P(A) + P(B)
When flipping a coin, the outcomes are mutually exclusive so only one of them can occur in any given flip. Since exactly one of them must occur and since they're mutually exclusive and equally likely to occur, the chance for a head is 1/2.

In a sample set composed of mutually-exclusive outcomes, the probability that at least one of those outcomes will occur equals 1, a sure event. So, the sum of the individual probabilities over all the outcomes equals 1. In the case of a coin flip, the sample space contains two events. The probability of each event is 1/2. The sum of the two probabilities equals the probability for either a head or a tail to occur, or 1.

Consider the rolling of a six-sided die. In this case, the sample space consists of six mutually-exclusive outcomes:

S = {1,2,3,4,5,6},
where

P(n) = 1/6, for n = 1, 2, ..., 6
When the probability for each outcome is equal (e.g. when flipping a coin or rolling one die), P(n) describes a uniform distribution. If you include a second die, the number of distinct outcomes increases to 36. That is, for each of the six outcomes of die #1 there are six possible combinations with die #2, with each of the pairs having an equal chance of occurring, namely 1/36.

To describe the results of rolling two dice, you could specify all 36 pairs. However, when rolling two dice, the sum of the pair usually interests the player more than the possible combinations. So, I will define the sample space for two dice as:

S = {2,3,4,5,6,7,8,9,10,11,12}
Each of the 11 events in this sample space consists of a set of pairs of mutually-exclusive outcomes. For example, the event of a roll yielding 4 consists of the following set of three pairs:

E(4) = {(1,3,), (3,1), (2,2)}
The probability for this event is the sum of the probabilities of its constituent (mutually-exclusive) pairs:

P(4) = 1/36 + 1/36 + 1/36 = 0.0833
Thus, the event of a roll yielding 7 occurs most often because its set of possible combinations contains the largest number of pairs. (In other words, there are more ways to get a 7 than any other number.) A roll yielding a 2 or a 12 (one pair each) occurs the least often. Since the number of pairs that comprise E(n), in this example, varies linearly as n varies from 2 to 7 or from 12 to 7, a plot of P(n) versus n, is triangular with its apex at n =7.

Density Distribution

The normal probability density distribution, F(x), describes the familiar symmetrical bell-shaped curve commonly associated with statistics.

F(X) = k1 * e-k2(x-µ)2

for - infinity <= x <= + infinity, where

The parameters m and s are the mean and standard deviation for this distribution. Since 99.7 percent of the area under this curve lies between x = m + 3s, where the value of F(x) is only 1.11 percent of its maximum, the domain of this function is often limited to these bounds in computer approximations.

Two distinctions need to be made between F(x) and the functions used by a digital computer to simulate a coin flip or a dice toss. F is a continuous function of the real variable, x, whereas a computer must use discrete functions of integers. Second, F is a probability density, the probability for events to occur per unit x. Instead of specifying the probability for a dart to hit a target at a point exactly x1 millimeters from the bullseye, F can be used to specify the chances for hitting the target over a finite range of x, by integrating F(x)dx over that range.