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Section 6.2 ~ Basics of Probability Introduction to Probability and Statistics Ms. Young

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Objective Sec. 6.2 After this section you will know how to find probabilities using theoretical and relative frequency methods and understand how to construct basic probability distributions.

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Basics of Probability Outcomes – the most basic possible results of observations or experiments Ex. ~ There are four outcomes of tossing two coins, HH, HT, TH, or TT Event – a collection of one or more outcomes that share a property of interest Ex. ~ Suppose you are only interested in the number of heads that appear when you toss two coins. There would be three events, 0 heads (TT), 1 head (HT or TH), or two heads (HH) Sec. 6.2

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Basics of Probability Cont’d… To express a probability, you use numbers between 0 and 1 inclusive A probability of 0 would represent an event that is impossible Ex. ~ Meeting a married bachelor A probability of 1 would represent an event that is certain to occur Ex. ~ Death and taxes! The probability of an event is written as P(event) Ex. ~ The probability of landing a head on a coin toss would be written as P(H) =.5 The scale to the right shows common expressions used to represent probabilities based on their level in comparison to 0 and 1 Ex. ~ A probability of.95 indicates that an event is very likely to occur (95 out of 100 times) Ex. ~ A probability of.30 indicates that an event is unlikely to occur (30 out of 100 times) Ex. ~ A probability of.01 describes an event that is very unlikely to occur (1 out of 100 times) Sec. 6.2

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Theoretical Probabilities Theoretical probabilities are probabilities that deal with equally likely outcomes (i.e., tossing a fair coin, rolling a fair die, spinning a roulette wheel, etc.) Calculating theoretical probabilities: Step 1: Count the total number of possible outcomes Step 2: Among all the possible outcomes, count the number of ways the event of interest, A, can occur Step 3: Determine the probability, P(A), from Sec. 6.2

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Example 1 Suppose you select a person at random from a large group. What is the probability that the person has a birthday in July? Assume that there are 365 days in a year. Since all birthdays are equally likely, we can use the 3 step process for calculating theoretical probabilities: Step 1: Each possible birthday represents an outcome, so there are 365 possible outcomes Step 2: July has 31 days, so 31 of the 365 possible outcomes represent the event of a July birthday Step 3: The probability that a randomly selected person has a birthday in July is Sec. 6.2

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Theoretical Probabilities Cont’d… Counting Outcomes – the total number of outcomes can be found by raising the individual outcome to the number of processes Ex. ~ What is the total number of outcomes of tossing two coins? Each coin has 2 outcomes (H or T), and there are 2 coins (2 tosses or processes), so there are 4 possible outcomes (2 2 = 4) when tossing two coins Ex. ~ What is the total number of outcomes of tossing three coins? Each coin has 2 outcomes (H or T), and there are 3 coins (3 tosses or processes), so there are 8 possible outcomes (2 3 = 8) when tossing three coins Sec. 6.2

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Example 2 Sec. 6.2 How many outcomes are there if you roll a fair die and toss a fair coin? The first process, rolling a fair die, has six possible outcomes, 1, 2, 3, 4, 5, or 6, and the second process, tossing a fair coin, has two possible outcomes (H or T). The total number of outcomes would be 12 ( 6 1 × 2 1 ). What is the probability of rolling two 1’s (snake eyes) when two fair dice are rolled? Rolling a single die has 6 equally likely outcomes, so rolling two dice has a total of 36 outcomes (6 2 = 36). Of the 36 outcomes, the event of interest (two 1’s) can only occur one way, so the probability of rolling two 1’s is

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Example 3 Sec. 6.2 What is the probability that in a randomly selected family with three children, the oldest child is a boy, the second child is a girl, and the youngest child is a girl? Assume that having boys and girls is equally likely. There are two possible outcomes for each birth: boy or girl For a family with three children, there would be 8 possible outcomes (2 3 = 8) BBB, BBG, BGG, GBB, GBG, GGB, GGG The probability of the birth order being BGG is

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Relative Frequency Probabilities Another method to determine probabilities is to approximate the probability of an event, A, occurring. This is known as the relative frequency (or empirical) method. Ex. ~ If we observe that it rains an average of 100 days per year, we can estimate the probability of it raining on a randomly selected day to be approximately.274 (100/365) Here is the general rule for this method: Step 1: Repeat or observe a process many times and count the number of times the event of interest, A, occurs. Step 2: Estimate P(A) by Sec. 6.2

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Example 4 Sec. 6.2 Geological records indicate that a river has crested above a particular high flood level four times in the past 2,000 years. What is the relative frequency probability that the river will crest above the high flood level next year? Because a flood of this magnitude occurs on average once every 500 years, it is called a “500-year flood.” The probability of having a flood of this magnitude in any given year is 1/500, or 0.002.

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Subjective Probabilities & Summary of Different Methods of Finding Probabilities A third method for determining probabilities is to estimate a subjective probability using experience or intuition Ex. ~ You could make a subjective estimate of the probability that a friend will be married in the next year or the probability that getting a good grade in statistics will help you get the job that you want Three approaches to Finding Probability Theoretical probability – when all outcomes are equally likely, divide the number of ways an event can occur by the total number of outcomes Relative frequency probability – based on observations or experiments. Divide the number of times the event occurred by the total number of observations Subjective probability – estimating based on experience or intuition Sec. 6.2

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Example 5 Sec. 6.2 Identify the method that resulted in the following statements. a. The chance that you will get married in the next year is zero. Subjective because it’s based on a feeling b. Based on government data, the chance of dying in an automobile accident is 1 in 7,000 (per year). Relative frequency probability because it’s based on observations on passed automobile accidents c. The chance of rolling a 7 with a twelve-sided die is 1/12. Theoretical probability because it is based on assuming that a fair twelve sided die is equally likely to land on any of its twelve sides.

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Probability of an Event Not Occurring Sometimes you might be interested in finding the probability that a particular event or outcome does not occur Ex. ~ The probability of a wrong answer on a multiple choice question with five possible answers The probability of answering it correctly would be.2 (1/5), so the probability of not answering it correctly would be.8 (4/5) The complement of an event, A, expressed as, consists of all outcomes in which A does not occur. The sum of the probabilities of A and must be 1, so the probability of can be given by Sec. 6.2

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Example 6 In a grocery store the scanning system was successful 384 out of 419 times. What is the probability that the scanner will not work? Sec. 6.2

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Probability Distributions A probability distribution is a visual display of the probabilities of certain events occurring in the form of a table or a histogram Ex. ~ Suppose you toss two coins simultaneously. Because each coin can land one of two ways (H or T), there are 4 possible outcomes (HH, TT, HT, & TH). The following table represents the outcomes and probabilities: Out of the 4 outcomes, there are 3 events, 2 heads (HH), 1 head (HT or TH), and 0 heads (TT). These probabilities result in a probability distribution which can represented as a table or a histogram: Sec. 6.2

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Probability Distributions Steps to making a probability distribution: Step 1: List all possible outcomes Step 2: Identify outcomes that represent the same event. Find the probability of each event. Step 3: Make a table or a histogram in which one column (or x-axis) represents the events and the other column (or y-axis) represents the probability Example 7: Make a probability distribution table for the number of heads that occur when three coins are tossed simultaneously. Step 1: List all possible outcomes Since there are 3 coins, there are a total of 8 outcomes (2 3 = 8): HHH, HHT, HTH, HTT, THH, THT, TTH, and TTT (refer to p.241 to see how these outcomes were constructed) Step 2: Identify outcomes that represent the same event. Find the probability of each. Since we are interested in the number of heads that occur, there would be 4 events, 0 heads (1/8 =.125), 1 head (3/8 =.375), 2 heads (3/8 =.375), or 3 heads (1/8 =.125) Sec. 6.2

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Probability Distributions Example 7 Cont’d: Make a probability distribution table for the number of heads that occur when three coins are tossed simultaneously. Step 3: Make a table Sec. 6.2

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