1. Chapter 4 Probability 4-1 Review and Preview 4-2 Basic Concepts of Probability 4-3 Addition Rule 4-4 Multiplication Rule: Basics 4-5 Multiplication Rule: Complements and Conditional Probability 4-6 Probabilities Through Simulations 4-7 Counting

2. Section 4-1 Review and Preview

3. Preview Rare Event Rule for Inferential Statistics: If, under a given assumption, the probability of a particular observed event is extremely small, we conclude that the assumption is probably not correct. Statisticians use the rare event rule for inferential statistics.

4. Section 4-2 Basic Concepts of Probability

5. Key Concept This section presents three approaches to finding the probability of an event. The most important objective of this section is to learn how to interpret probability values.

6. Part 1 Basics of Probability

7. Events and Sample Space  Event any collection of results or outcomes of a procedure  Simple Event an outcome or an event that cannot be further broken down into simpler components  Sample Space for a procedure consists of all possible simple events; that is, the sample space consists of all outcomes that cannot be broken down any further

8. Events and Sample Space We flip a coin once: – Example of an event is 1 head (H) – Simple event is (H) cannot be broken down any further. – Complete Sample Space is {H, T}

9. Events and Sample Space We flip a coin three times: – Example of an event is 2 heads and one tail (H,H,T) – Simple event is (H,H,T) cannot be broken down any further. – Complete Sample Space is {HHH, HHT, HTH, HTT, THT, TTH, THH, TTT} all of the possible outcomes

10. Notation for Probabilities P - denotes a probability. A, B, and C - denote specific events. P(A) - denotes the probability of event A occurring.

11. Basic Rules for Computing Probability Rule 1: Relative Frequency Approximation of Probability Conduct (or observe) a procedure, and count the number of times event A actually occurs. Based on these actual results, P(A) is approximated as follows: P(A) =P(A) = # of times A occurred # of times procedure was repeated

12. Basic Rules for Computing Probability - continued Rule 2: Classical Approach to Probability (Requires Equally Likely Outcomes) Assume that a given procedure has n different simple events and that each of those simple events has an equal chance of occurring. If event A can occur in s of these n ways, then P(A) = number of ways A can occur number of different simple events s n =

13. Basic Rules for Computing Probability - continued Rule 3: Subjective Probabilities P(A), the probability of event A, is estimated by using knowledge of the relevant circumstances.

14. Computing Probability What is the likelihood of getting 2 heads and 1 tail when a coin is flipped three times? How many ways can you get 2 H and 1 T? How many events in the total sample space? The probability is # possible ways/total # of ways.

15. Law of Large Numbers As a procedure is repeated again and again, the relative frequency probability of an event tends to approach the actual probability.

16. Probability Limits  The probability of an event that is certain to occur is 1.  The probability of an impossible event is 0.  For any event A, the probability of A is between 0 and 1 inclusive. That is, 0  P(A)  1. Always express a probability as a fraction or decimal number between 0 and 1.

17. Possible Values for Probabilities

18. Complementary Events The complement of event A, denoted by A, consists of all outcomes in which the event A does not occur.

19. Rounding Off Probabilities When expressing the value of a probability, either give the exact fraction or decimal or round off final decimal results to three significant digits. (Suggestion: When a probability is not a simple fraction such as 2/3 or 5/9, express it as a decimal so that the number can be better understood.)

20. Interpreting Probability Based on recent results, the probability of someone in the US being injured while using sports or recreation equipment is 1/500. What does it mean when we say that the probability is 1/500? Is such an injury unusual?

21. Interpreting Probability When predicting the chance that we will elect a Republican President in the year 2014, we could reason that there are two possible outcome (Rep and not Rep), so the probability of a Republican President is ½ or 0.5. Is this reasoning correct? Why or why not?

22. Probability Express the indicated degree of likelihood as a probability value between 0 and 1. – In one of Kansas’ instant lottery games, the chances of a win are stated as 4 in 21. – A WeatherBug forecast for Friday was stated as a “80% chance of rain.” – If you make a random guess for the answer to a true/false test question, there is a 50-50 chance of being correct.

23. Probability Express the indicated degree of likelihood as a probability value between 0 and 1. – When rolling a single die at the Venetian Casino in Las Vegas, what is the probability of rolling a “7.” – What are the chances of drawing 5 aces when selecting cards from a shuffled deck. – When randomly selecting a day of the week, you are certain to select a day containing the letter “y.”

24. Probability Which of the following values cannot be probabilities? 3:12/55/2-0.50.5 123/321321/12301

25. Probability Refer to our example of flipping a coin three times. – What is the probability of getting exactly one T? – What is the probability of getting exactly two T? – What is the probability of getting three T?

26. Probability The 110 th Congress of the United States included 84 male Senators and 16 female Senators. If one of these Senators is randomly selected, what is the probability that a woman is selected? Does this probability agree with a claim that men and women have the same chance of being elected as Senators?

27. Probability In a recent year, 281 of the 290,789,000 people in the US were struck by lightning. Estimate the probability that a randomly selected person in the US will be struck by lightning this year. Is a golfer reasoning correctly if he or she is caught out in a thunderstorm and does not seek shelter from lightning during a storm because the probability of being struck is so small?

28. Recap In this section we have discussed:  Rare event rule for inferential statistics.  Probability rules.  Law of large numbers.  Complementary events.  Rounding off probabilities.