Probability Models (Probability Central)

Sample
Space

In any number of given trials, the set of all possible outcomes is called the sample space (each possible outcome is a sample and the sample space contains all possible samples). The sample space is often presented by the letter S.

Example:
In tossing a coin, there are two possible outcomes. Therefore, the sample space is:

S = {heads , tails} or S = {H , T}

Example:
Grab 26 index cards and put each letter of the alphabet on each card. Then put all the cards into a bag and randomly take out a card. The sample space for this experiment is:

S = {a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z}
Or
S = {a…z}

Example:
When you roll a die, the sample space is:

S = {1,2,3,4,5,6}
Or
S = {1…6}

Event

The sample space lists the possible outcomes of a random phenomenon. We, however, must also give the probability with which these outcomes occur to complete the mathematical model for the random phenomenon. To do this, we must find probabilities of single outcomes as well as sets of outcomes.
Event: An event is the set of outcomes of a random phenomenon, that is, a subset of a sample space.

Example:

When asking four people to vote on a subject, assume the sample space, S, for all four votes is the list of all possible outcomes in the form of two affirmatives. Call this event A. Event A expressed as a set of outcomes is:

A = {YYNN , NNYY , YNYN , NYNY , YNNY , NYYN}

Event A, could easily be described as "exactly two affirmatives". This is a much shorter way of expressing event A. To even shorten this description, it could be said "A is the event that X=2, which we write as {X=2}. The event "2 or more affirmatives" can similarly be described as {X ³ 2}.
We can write the probability of event A as P(A). The description above is the basis for assigning probability of an event. In a sample space containing sample counts that are equally likely to occur, the probability of an event is:

                 P(A) = Count of outcomes in A
                      Count of outcomes in S

Example:

Find the probability of drawing a heart on a single random draw from a well shuffled deck of cards.

The number of outcomes in S is 52, of which 13 are hearts. Thus,

         P(Drawing a heart) ₌ Count of outcomes in A ₌ 13
                             Count of outcomes in S   52

Most of the time, the outcomes are not equally likely. Therefore, we need to know the general rule to figure out the probability of unequally likely outcomes.

Example:
The sample space of the colors of M&M's (a well-known colored candy) in a bag is:

S = {brown , green , orange , red , tan , yellow}

The probability of each outcome is determined by the total number of each color over the total number of M&M's produced.

Color	Brown	Red	Yellow	Green	Orange	Tan
Probability	.3	.2	.2	.1	.1	.1

Notice that the sum of all the probabilities is 1.