Section 6.2Probability Models

Sample Space

• The sample space S of a random phenomenon is the set of all possible outcomes.

• For a flipped coin, the sample space is S = {H, T}. For a child's sex it is S = {girl, boy}. For a table of random digits it is S = {0,1,2,3,4,5,6,7, 8,9}.

• For a coin flipped and a die rolled it is S = {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}. This may also be written S = {H1, T1, H2, T2, H3, T3, H4, T4, H5, T5, H6, T6}. A tree diagram may help you determine all of the possible outcomes.

Multiplication Principle

• If you can do one task in a number of ways and a second task in b number of ways, then both tasks can be done in a * b number of ways.

• If we wish to list the possible outcomes of the sex of 4 children, we would list S = {GGGG, GGGB, GGBG, GBGG, BGGG, ..., BBBB}. Note that there would be 2x2x2x2 = 16 possible outcomes. But, if we simply wanted to know the number of girls, we would have a simpler sample space, S = {0, 1, 2, 3, 4}.

“Event”

• An event is an outcome or a set of outcomes of a random phenomenon. So, an event is a subset of the sample space.

Probability Rules

1. The probability P (A) of any event A satisfies 0 <= P (A) <= 1. Any probability is a number between 0 and 1.

2. If S is the sample space of a probability model, then P (S) = 1. All probabilities together must have a probability of 1.

Probability Rules

3. The complement of any event A is the event that A does not occur, written as AC. The complement rule states that P (AC) = 1 - P (A). The probability that an event does not occur is 1 minus the probability that the event does occur.

4. Two events A and B are disjoint if they have no outcomes in common and so can never occur simultaneously. If A and B are disjoint, P (A or B) = P (A) + P (B). This is the addition rule.