Introduction to Probability

The sample space is the set of all possible outcomes in an experiment.

Example 1: If a die is rolled, the sample space S is given by

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

Example 2: If two coins are tossed, the sample space S is given by

S = {HH,HT,TH,TT} , where H = head and T = tail.

Example 3: If two dice are rolled, the sample space S is given by

S = { (1,1),(1,2),(1,3),(1,4),(1,5),(1,6)
         (2,1),(2,2),(2,3),(2,4),(2,5),(2,6)
         (3,1),(3,2),(3,3),(3,4),(3,5),(3,6)
         (4,1),(4,2),(4,3),(4,4),(4,5),(4,6)
         (5,1),(5,2),(5,3),(5,4),(5,5),(5,6)
         (6,1),(6,2),(6,3),(6,4),(6,5),(6,6) }

We define an event as some specific outcome of an experiment. An event is asubset of the sample space.

Example 4: A die is rolled (see example 1 above for the sample space). Let us define event E as the set of possible outcomes where the number on the face of the die is even. Event E is given by

E = {2,4,6}

Example 5: Two coins are tossed (see example 2 above for the sample space). Let us define event E as the set of possible outcomes where the number of head obtained is equal to two. Event E is given by

E = {(HT),(TH)}

Example 6: Two dice are rolled (see example 3 above for the sample space). Let us define event E as the set of possible outcomes where the sum of the numbers on the faces of the two dice is equal to four. Event E is given by

E = {(1,3),(2,2),(3,1)}

How to Calculate Probabilities?

1 - Classical Probability Formula: It is based on the fact that all outcomes are equally likely.

	Total number of outcomes in E
P(E)=	________________________________________________
	Total number of outcomes in the sample space

Example 7: A die is rolled, find the probability of getting a 3.

The event of interest is "getting a 3". so E = {3}.

The sample space S is given by S = {1,2,3,4,5,6}.

The number of possibel outcomes in E is 1 and the number of possible outcomes in S is 6. Hence the probability of getting a 3 is P("3") = 1 / 6.

Example 8: A die is rolled, find the probability of getting an even number.

The event of interest is "getting an even number". so E = {2,4,6}, the even numbers on a die.

The sample space S is given by S = {1,2,3,4,5,6}.

The number of possible outcomes in E is 3 and the number of possible outcomes in S is 6. Hence the probability of getting a 3 is P("3") = 3 / 6 = 1 / 2.

2 - Empirical Probability Formula: It uses real data on present situations to determine how likely outcomes will occur in the future. Let us clarify this using an example

30 people were asked about the colors they like and here are the results: