Probability Models Section 6.2

The Language of Probability What is random? What is random? Empirical means that it is based on observation rather than theorizing. Empirical means that it is based on observation rather than theorizing. Probability describes what happens in MANY trials. Probability describes what happens in MANY trials. Example 6.9: Long-term relative frequency Example 6.9: Long-term relative frequency

Randomness and Probability We call a phenomenon random if individual outcomes are uncertain but there is nonetheless a regular distribution of outcomes in a large number of repetitions. We call a phenomenon random if individual outcomes are uncertain but there is nonetheless a regular distribution of outcomes in a large number of repetitions. The probability of any outcome of a random phenomenon is the proportion of times the outcome would occur in a very long series of repetitions. That is, probability is long-term relative frequency. The probability of any outcome of a random phenomenon is the proportion of times the outcome would occur in a very long series of repetitions. That is, probability is long-term relative frequency.

Assignment Page 410 exercises 6.21 – 6.28 Page 410 exercises 6.21 – 6.28

Toss a coin… We cannot know the outcome in advance. We cannot know the outcome in advance. The outcome will be either heads or tails. The outcome will be either heads or tails. Each of these outcomes has the probability of ½. Each of these outcomes has the probability of ½. The basis of all probability models is a list of all possible outcomes and a probability for each outcome. The basis of all probability models is a list of all possible outcomes and a probability for each outcome.

Sample Spaces The sample space S of a random phenomenon is the set of all possible outcomes. The sample space S of a random phenomenon is the set of all possible outcomes. To specify S, we must state what constitutes an individual outcome and then state which outcomes can occur. To specify S, we must state what constitutes an individual outcome and then state which outcomes can occur.

How to count! Being able to properly enumerate the outcomes in a sample space will be critical to determining probabilities. Being able to properly enumerate the outcomes in a sample space will be critical to determining probabilities. Two techniques are very helpful in making sure you don’t accidentally overlook any outcomes. Two techniques are very helpful in making sure you don’t accidentally overlook any outcomes. These techniques are the tree diagram and the multiplication principle. These techniques are the tree diagram and the multiplication principle.

Tree Diagram Toss a coin H T Roll a die H1 H2 H3 H4 H5 H6 T1 T2 T3 T4 T5 T6

Multiplication Principle If you can do one task in n number of ways and a second task in m number of ways, then both tasks can be done in nXm number of ways. If you can do one task in n number of ways and a second task in m number of ways, then both tasks can be done in nXm number of ways.

Nondiscrete sample space Some sample spaces are simply too large to allow all of the possible outcomes to be listed. Some sample spaces are simply too large to allow all of the possible outcomes to be listed. Example 6.12 Example 6.12

With and Without Replacement Sampling with replacement means that once you’ve made your first selection, you return it so that it can be chosen again. Sampling with replacement means that once you’ve made your first selection, you return it so that it can be chosen again. Sampling without replacement means that you do not return your first selection. Sampling without replacement means that you do not return your first selection.

Assignment Page 416, problems 6.29 – 6.36 Page 416, problems 6.29 – 6.36

Probability of an Event The probability of event A is the number of ways event A can occur divided by the total number of possible outcomes. The probability of event A is the number of ways event A can occur divided by the total number of possible outcomes. P(A) = The Number Of Ways Event A Can Occur P(A) = The Number Of Ways Event A Can Occur The Total Number Of Possible Outcomes The Total Number Of Possible Outcomes

A pair of dice is rolled, one black and one white. Find the probability of each of the following events. 1.The total is The total is at least The total is less than The total is at most The total is 7. 6.The total is 2. 7.The total is The numbers are 2 and 5. 9.The black die has 2 and the white die has The black die has 2 or the white die has 5.

Probability Rules All probabilities are between 0 and 1 inclusive All probabilities are between 0 and 1 inclusive The sum of all the probabilities in the sample space is 1 The sum of all the probabilities in the sample space is 1 The probability of an event which cannot occur is 0. The probability of an event which cannot occur is 0. The probability of any event which is not in the sample space is zero. The probability of any event which is not in the sample space is zero. The probability of an event which must occur is 1. The probability of an event which must occur is 1. The probability of an event not occurring is one minus the probability of it occurring. The probability of an event not occurring is one minus the probability of it occurring. P(E') = 1 - P(E)

The Addition Rule Two events are disjoint (mutually exclusive) if they have no outcomes in common. Two events are disjoint (mutually exclusive) if they have no outcomes in common. If two events are disjoint, the number of ways one or the other can occur is If two events are disjoint, the number of ways one or the other can occur is

Set Notation Union Union Empty Event Empty Event Intersect Intersect

Examples 6.13 Complement Rule 6.13 Complement Rule 6.14 Applying Probability Rules 6.14 Applying Probability Rules 6.15 Applying Probability Rules 6.15 Applying Probability Rules 6.16 Applying Probability Rules 6.16 Applying Probability Rules

Assignment Page 423, problems 6.37 – 6.44 Page 423, problems 6.37 – 6.44

Independence and the multiplication rule Two events A and B are independent if knowing that one occurs does not change the probability that the other occurs. Two events A and B are independent if knowing that one occurs does not change the probability that the other occurs. If A and B are independent, If A and B are independent,

Independent or not independent? Example 6.17 Example 6.17 Example 6.18 Example 6.18

Applying the Multiplication Rule Example 6.19 Example 6.19

Independence and the Complement Rule Example 6.21 Example 6.21

Assignment Page 430, exercises 6.45 – 6.52 Page 430, exercises 6.45 – 6.52

Section Exercises Page 432, exercises 6.53 – 6.63 Page 432, exercises 6.53 – 6.63