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Sta220 - Statistics Mr. Smith Room 310 Class #7 • Questions about the project? Where We’ve Been • Identified the objective of inferential statistics: to make inferences about a population on the basis of information in a sample • Introduced graphical and numerical descriptive measure for both quantitative and qualitative data. Where We’re Going • Develop probability as a measure of uncertainty • Introduce basic rules for finding probabilities • Use a probability as a measure of reliability for an inference • Use a probability in random sampling Section 3.1 An experiment is an act or process of observation that leads to a single outcome that cannot be predicted with certainty. A sample point is the most basic outcome of an experiment. Example: Listing Sample Point for a Coin-Tossing Experiment Two coins are tossed, and their up faces are recorded. List all sample points for this experiment. Figure 3.1 Tree diagram for the coin-tossing experiment Copyright © 2013 Pearson Education, Inc.. All rights reserved. Figure 3.2 Venn diagrams for the three experiments from Table 3.1 Copyright © 2013 Pearson Education, Inc.. All rights reserved. The sample space of an experiment is the collection of all its sample points. Table 3.1 Copyright © 2013 Pearson Education, Inc.. All rights reserved. The probability of sample point is a number between 0 and 1 which measures the likelihood that the outcome will occur when the experiment is performed. Probability Notation • Probabilities can be expressed as a fraction (always reduced), decimal, or percent. • P(A) = 0.123 means the probability of event A is 0.123 • Unless noted, round probabilities to three decimal places. Law of Large Numbers states the relative frequency of the number of times that an outcome occurs when an experiment is replicated over and over again approaches the true probability of the outcome. Figure 3.3 Proportion of heads in N tosses of a coin Copyright © 2013 Pearson Education, Inc.. All rights reserved. Procedure Copyright © 2013 Pearson Education, Inc.. All rights reserved. EXAMPLE A Probability Model In a bag of peanut M&M milk chocolate candies, the colors of the candies can be brown, yellow, red, blue, orange, or green. Suppose that a candy is randomly selected from a bag. The table shows each color and the probability of drawing that color. Verify this is a probability model. Color Probability Brown 0.12 Yellow 0.15 Red 0.12 Blue 0.23 Orange 0.23 Green 0.15 •All probabilities are between 0 and 1, inclusive. • Because 0.12 + 0.15 + 0.12 + 0.23 + 0.23 + 0.15 = 1, rule 2 (sum of all probabilities must equal 1) is satisfied. Example: Die Tossing Experiment A fair die is tossed, and the up face is observed. If the face is even, you win $1. Otherwise, you lose $1. What is the probability that you win? Example: Die Tossing Experiment The sample space is S: {1, 2, 3, 4, 5, 6} Since the die is balanced, we assign a probability of 1/6 to each sample points in the sample space. 1 1 1 1 𝑃 𝐴 = + + = 6 6 6 2 Figure 3.5 Die-toss experiment with event A, observe an even number Copyright © 2013 Pearson Education, Inc.. All rights reserved. An event is a specific collection of sample points. Example: Probability of an EventCoin-Tossing Experiment Consider the experiment of tossing two unbalanced coins. Because the coins are not balanced, their outcomes ( H or T) are not equiprobable. Suppose the correct probabilities associated with the sample points are given in the accompanying table. [Note: The necessary for assigning the probabilities to sample points are satisfied.] Consider the event A: {Observe exactly one head} B: {Observe at least one head} Calculate the probability of A and the probability of B. Sample Point Probability HH 4/9 HT 2/9 TH 2/9 TT 1/9 2 2 4 𝑃 𝐴 = 𝑃 𝐻𝑇 + 𝑃 𝑇𝐻 = + = 9 9 9 4 So 𝑃 𝐴 = 9 4 2 2 𝑃 𝐵 = 𝑃 𝐻𝐻 + 𝑃 𝐻𝑇 + 𝑃 𝑇𝐻 = + + 9 9 9 8 = 9 So 𝑃(𝐵) = 8/9 Probability of an Event The probability of an event A is calculated by summing the probabilities of the sample points in the sample space for A. Steps for Calculating Probabilities of Events 1. Define the experiment: that is, describe the process used to make an observation and the type of observation that will be recorded. 2. List of the sample points. 3. Assign probabilities to the sample points. 4. Determine the collection of sample points contained in the even of interest. 5. Sum the sample point probabilities to get the probability of the event. Combinations Rule Example: Selecting 5 Movies from Suppose a movie reviewer for a newspaper reviews 5 movies each month. This month, the reviewer has 20 new movies from which to make the selection. How many different samples of 5 movies can be selected from the 20? N = 20 and n = 5 20 20! 20! = = 5 5! 20 − 5 ! 5! 15! 20 ∙ 19 ∙ 18 ∙ ⋯ ∙ 3 ∙ 2 ∙ 1 = 5 ∙ 4 ∙ 3 ∙ 2 ∙ 1 15 ∙ 14 ∙ 13 … 3 ∙ 2 ∙ 1 20 ∙ 19 ∙ 18 ∙ 17 ∙ 16 = 5∙4∙3∙2∙1 = 15,504