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Chapter 4 Created by Laura Ralston 4-1 4-2 4-3 4-4 Classical Probability Probability Rules Basic Counting Rules Additional Counting Techniques Probability ◦ What is it? ◦ How is it computed in a variety of important circumstances? ◦ How is it useful to me? From the time you awake until you go to bed, you make decisions regarding the possible events that are governed at least in part by chance. Should I carry an umbrella today? Should I accept that new job? Will I have enough gas to get to school? In general, probability is defined as the likelihood of an event occurring ◦ Card Games ◦ Slot Machines ◦ Lotteries In theory, probability is the underlying foundation on which inferential statistics is built ◦ Insurance ◦ Investments ◦ Weather forecasting Objectives: ◦ Identify the sample space of a probability event ◦ Calculate basic probabilities Probability experiments: any process in which the result is random in nature ◦ Flip a Coin ◦ Roll a die ◦ Answer multiple choice questions with 4 possibilities ◦ Take a pregnancy test ◦ Predict grade in MATH 2200 for Fall 2008 Outcome: the individual result of a trial in a probability experiment Sample Space: the set of ALL possible outcomes of a probability experiment In the previous examples, the sample spaces were found by observation or reasoning, BUT what if the probability experiment is “more complex”? ◦ Roll TWO dice ◦ Gender of children if a family has 3 children ◦ Select card from a standard 52-card deck If the probability experiment is “more complex”, we can use a ◦ Two-way Table ◦ Tree Diagram: device that organizes the outcomes of a probability experiment with several stages— tree begins with outcomes for first stage and branches for each additional possibility 1) When ordering a pizza with a coupon, you can have a choice of crusts: thin (T), handtossed (H), or stuffed (S). You can also choose one topping from the following: pepperoni (P), ham (M), sausage (G), onion (O), bell pepper (B), or olives (V) 2) A family has three children. Give the sample space in regard to the sex of the children Simple Event: an event with one outcome ◦ Roll a die and a 6 shows ◦ Flip a coin and a HEAD shows ◦ Take a pregnancy test and a NEGATIVE result shows Compound Event: an event with two or more outcomes ◦ Roll a die and an odd number shows (1,3, or 5) ◦ Answer a multiple choice question with 4 possibilities (a, b, c, or d) ◦ Predict grade in MATH 2200 (A, B, C, D, or F) P: denotes a probability A, B, and C: denotes a specific event P(A): is read “the probability of event A” Classical Probability (aka Theoretical Probability) Empirical Probability Subjective Probability Uses a probability value based on an educated guess or estimate, employing opinions and inexact information ◦ Weather Prediction ◦ Earthquake Prediction ◦ Braves win pennant in 2011 Prediction As a probability experiment is repeated again and again, the relative frequency probability of an event tends to approach the actual probability ◦ Conduct experiment: Flip coin 10 times per individual and record number of times a HEAD occurs First type of probability studied in 17th-18th centuries Most precise type of probability Assumes that all outcomes in the sample space are equally likely to occur Final results can be expressed as fractions, decimals, or percentages ◦ Always simply fractions ◦ Round decimals to three places. If extremely small, it is okay to round the decimal to the first nonzero digit. 3) If a die is rolled one time, find the probability: ◦ a. P(getting a 4) ◦ b. P(getting an even number) ◦ c. P(getting a number greater than 3 or an odd number) 4) A couple has three children. Find the probability: ◦ a. P(all boys) ◦ b. P(exactly two boys or two girls) ◦ c. P(at least one child of each gender) In a college class of 250 graduating seniors, 50 have jobs waiting, 10 are going to medical school, 20 are going to law school, and 80 are going to various other kinds of graduate schools. ◦ 5) How many have no jobs or are not attending graduate school? ◦ 6) Select one graduate at random. Find the probability: a. P(student is going to graduate school) b. P(student will have to start paying back loans after 6 months---does not continue in school)