Probability and Its Postulates

We consider three definitions of probability
1.    Classical probability
2.    Relative frequency probability
3.    Subjective probability

Classical Probability

Classical probability is the proportion of times that an event will occur, assuming that all outcomes in a sample space are equally to occur. The probability of an event A is

where NA is the number of of outcomes that satisfy the condition of event A  and N is the total number of outcomes in the sample space.

Example: Karlyn Akimoto operates a small computer store. On a particular day she has three Gateway and two Compaq computers in stock. Suppose that Susan Spencer comes into the store to purchase two computers. Susan is not concerned about which brand she purchases—they all have the same operating specifications—so Susan selects the computers purely by chance: Any computer on the shelf is equally likely to be selected. What is the probability that Susan will purchase one Gateway and one Compaq computer?

Solution: Te answer can be obtained using classical probability. To begin, the sample space is defined as all possible pairs of two computers that can be selected from the store. The number of pairs is then counted, as is the number of outcomes that meet the condition—one Gateway and one Compaq. Define the three Gateway computers as, G1, G2, and G3 and two Compaq computers as C1 and C2. The sample space, S, contains the following pairs of computers:

         S = {G1C1, G1C2, G2C1, G2,C2, G3,C1, G3C2, G1G2, G1G3, G2G3, C1C2}

The number of computers in the sample space is 10. If A is the event “One Gateway and one Compaq computer are chosen,” then the number, NA, of outcomes that have one Gateway and one Compaq computer is 6. Therefore, the required probability of event A—one Gateway and one Compaq—is

Formula for Determining the Number of Combinations

The counting process can be generalized by using the following equation to compute the number of combinations of n items taken k at a time:

Thus the number of combinations of the five computers taken two at a time is the number of elements in the sample space:

Example: Suppose that Karlyn’s store now contains 10 Gateway computers, 5 Compaq Computers, and 5 Acer computers. Susan enters the store and wants to purchase 3 computers. The computers are selected purely by chance from the shelf. Now what is the probability 2 Gateway computers and 1 Compaq computer are selected?

 Solution: The classical definition of probability will be used. But in this example the combinations formula will be used to determine the number of outcomes in the sample space and the number of outcomes that satisfy the condition: [2 Gateways and 1 Compaq].

The total number of outcomes in the sample space is

The number of ways that we can select 2 Gateway computers from the 10 available is computed by

Similarly, the number of ways that we can select 1 Compaq computer from the 5 available is computed by

Therefore, the number of outcomes that satisfy event A is

Finally, the probability of A = [2 Gateways and 1 Compaq] is