Probability Models. Section 6.2

Transcription

1 Probability Models Section 6.2

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

3 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. 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.

4 Assignment Page 410 exercises

5 Toss a coin We cannot know the outcome in advance. The outcome will be either heads or tails. 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.

6 Sample Spaces 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.

7 How to count! 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. These techniques are the tree diagram and the multiplication principle.

8 Tree Diagram 1 H1 2 H2 Toss a coin H Roll a die H3 H4 H5 H6 T T1 T2 T3 T4 T5 T6

9 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.

10 Nondiscrete sample space Some sample spaces are simply too large to allow all of the possible outcomes to be listed. Generate a random decimal number between 0 and 1. Nondiscrete sample spaces

11 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 without replacement means that you do not return your first selection.

12 Assignment Page 416, problems

13 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. P(A) = The Number Of Ways Event A Can Occur The Total Number Of Possible Outcomes

14 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 The total is The total is The numbers are 2 and The black die has 2 and the white die has The black die has 2 or the white die has

15 Probability Rules All probabilities are between 0 and 1 inclusive 0 PE ( ) 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 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 not occurring is one minus the probability of it occurring. P(E') = 1 - P(E)

16 The Addition Rule 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 n( A or B) n( A) n( B)

17 Set Notation Union Empty Event Intersect

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

19 Assignment Page 423, problems

20 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. If A and B are independent, n( A and B) n( A) n( B)

21 Example 6.17 Example 6.18 Independent or not independent?

22 Applying the Multiplication Rule Example 6.19

23 Independence and the Example 6.21 Complement Rule

24 Assignment Page 430, exercises

25 Section Exercises Page 432, exercises