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##### Probability

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**Probability**Chapter 4**Basic Probability**We are going to study probability so that we can use it later during the study of inferential statistics. 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.**Basic Probability**• An event is any collection of results or outcomes of a procedure. • A simple event is an outcome or an event that cannot be further broken down into simpler components. • The 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.**Basic Probability**Flipping a coin Let h denote heads and t denote tails**Basic Probability**Flipping a coin Let h denote heads and t denote tails**Basic Probability**Flipping a coin Let h denote heads and t denote tails**Basic Probability**Flipping a coin Let h denote heads and t denote tails**Basic Probability**Flipping a coin Let h denote heads and t denote tails**Basic Probability**Flipping a coin Let h denote heads and t denote tails**Basic Probability**Flipping a coin Let h denote heads and t denote tails**Basic Probability**Notation for Probabilities • P denotes a probability**Basic Probability**Notation for Probabilities • P denotes a probability • A, B, and C denote specific events.**Basic Probability**Notation for Probabilities • P denotes a probability • A, B, and C denote specific events. • P(A) denotes the probability of an event A occurring.**Basic Probability**Notation for Probabilities • P denotes a probability • A, B, and C denote specific events. • P(A) denotes the probability of an event A occurring. Ex. When flipping a coin; if A={heads}, then P(A)=0.5**Basic Probability**Finding Probabilities**Basic Probability**Finding Probabilities 1. Relative Frequency Approximation of Probability Conduct or observe a procedure, and count the number of times that event Aactually occurs. Based on these actual results,P(A) is approximated as follows:**Basic Probability**Finding Probabilities 1. Relative Frequency Approximation of Probability Conduct or observe a procedure, and count the number of times that event Aactually occurs. Based on these actual results,P(A) is approximated as follows:**Basic Probability**Finding Probabilities 2.Classical Approach to Probability (Requires Equal Likely Outcomes) Assume that a given procedure has ndifferent simple events and that each of those simple events has an equal chance of occurring. If event A can occur in sof these nways, then:**Basic Probability**Finding Probabilities 2. Classical Approach to Probability (Requires Equal Likely Outcomes) Assume that a given procedure has ndifferent simple events and that each of those simple events has an equal chance of occurring. If event A can occur in sof these nways, then:**Basic Probability**Finding Probabilities 2. Classical Approach to Probability (Requires Equal Likely Outcomes) Assume that a given procedure has ndifferent simple events and that each of those simple events has an equal chance of occurring. If event A can occur in sof these nways, then: 3. Subjective Probabilities P(A), the probability of event A is estimated by using knowledge of the relevant circumstances**Basic Probability**• Law of Large Numbers tells us that relative frequency approximations tend to get better with more observations.**Basic Probability**• Law of Large Numbers tells us that relative frequency approximations tend to get better with more observations. • Probability and Outcomes That are not Equally Likely Don’t assume outcomes are equally as likely when you know nothing about the likelihood of each outcome.**Basic Probability**Ex. Given that there were 6,511,100 cars that crashed among 135,670,000 registered cars in the U.S. find the Probability that a random car will crash this year in the U.S.**Basic Probability**Ex. Given that there were 6,511,100 cars that crashed among 135,670,000 registered cars in the U.S. find the Probability that a random car will crash this year in the U.S.**Basic Probability**Ex. When studying the affect of heredity on height, we can express each individual genotype, AA, Aa, aA, and aa, on an index card and shuffle the four cards and randomly select one of them. What is the probability that we select a genotype in which the two components are different?**Basic Probability**Ex. When studying the affect of heredity on height, we can express each individual genotype, AA, Aa, aA, and aa, on an index card and shuffle the four cards and randomly select one of them. What is the probability that we select a genotype in which the two components are different?**Basic Probability**Ex. Given that the population of Alaska is 0.2% of the total U.S. population, find the probability that the next President of the United States is from Alaska.**Basic Probability**Ex. Given that the population of Alaska is 0.2% of the total U.S. population, find the probability that the next President of the United States is from Alaska. The since Alaska’s remoteness presents a challenge to politicians there.**Basic Probability**Ex. What is the probability of Drawing a King from a standard deck of 52 cards?**Basic Probability**Ex. What is the probability of Drawing a King from a standard deck of 52 cards?**Basic Probability**Ex. Find the probability that when a couple has 3 kids, that exactly 1 will be a girl. Assume that boys and girls are equally as likely for each birth.**Basic Probability**Ex. Find the probability that when a couple has 3 kids, that exactly 1 will be a girl. Assume that boys and girls are equally as likely for each birth.**Basic Probability**Ex. When Mendel conducted his famous genetics experiments with peas, one sample of offspring consisted of 428 green peas and 152 yellow peas. Based on those results, estimate the probability of getting an offspring pea that is green.**Basic Probability**Ex. When Mendel conducted his famous genetics experiments with peas, one sample of offspring consisted of 428 green peas and 152 yellow peas. Based on those results, estimate the probability of getting an offspring pea that is green.**Basic Probability**• The probability of an impossible event is 0. • The probability of an event that is certain to occur is 1. • For any event A, the probability of A is between 0 and 1 inclusive. That is, . • The complement of event A,denoted by , consists of all outcomes in which event A does not occur.**Basic Probability**Ex. A typical question on an SAT test requires the test taker to select one of five possible choices: A, B, C, D, or E. The probability of correctly answering a question when guessing is 1/5 or 0.2 Find the probability of making a random guess and not being correct i.e. being incorrect.**Basic Probability**Ex. A typical question on an SAT test requires the test taker to select one of five possible choices: A, B, C, D, or E. The probability of correctly answering a question when guessing is 1/5 or 0.2 Find the probability of making a random guess and not being correct i.e. being incorrect.**Basic Probability**Rounding Probabilities Give either the exact fraction or decimal or round off final decimal answer to three significant digits. • 0.04799219 0.0480 • 1/3 1/3 or 0.333 • 2/4 ½ or 0.5 • 1941/3405 .570**Addition Rule**We are going to learn the addition rule for probabilities, which allows us to find the probabilities of events that can be expressed as P(A or B), which denote the probability of Aoccurring or Boccurring or both occurring.**Addition Rule**We are going to learn the addition rule for probabilities, which allows us to find the probabilities of events that can be expressed as P(A or B), which denote the probability of Aoccurring or Boccurring or both occurring. A compound event is any event combining two or more simple events.**Addition Rule**Consider the following table.**Addition Rule**Consider the following table. If 1 subject is randomly selected from the 98 subjects given a polygraph test, find the probability of selecting a subject who had a positive test result or lied.**Addition Rule**Consider the following table. If 1 subject is randomly selected from the 98 subjects given a polygraph test, find the probability of selecting a subject who had a positive test result or lied.**Addition Rule**Consider the following table. If 1 subject is randomly selected from the 98 subjects given a polygraph test, find the probability of selecting a subject who had a positive test result or lied.**Addition Rule**Formal Addition Rule where denotes the probability that Aand Bboth occur at the same time as an outcome. Intuitive Addition Rule To find , find the sum of the # of ways event Acan occur and the # of ways event B can occur, adding in such a way as to not double count. The is equal to the sum divided by the total number of outcomes**Addition Rule**Events AandBare disjoint ( or mutually exclusive) if they cannot occur at the same time. (That is disjoint events do not overlap.) Note: If AandBare disjoint, then**Addition Rule**Events AandBare disjoint ( or mutually exclusive) if they cannot occur at the same time. (That is disjoint events do not overlap.) Note: If AandBare disjoint, then Complementary Events**Addition Rule**FBI data show that 62.4% of murders are cleared by arrests. We can express the probability of a murder being cleared by an arrest as P(cleared)=0.624. For a randomly selected murder, find P(**Addition Rule**FBI data show that 62.4% of murders are cleared by arrests. We can express the probability of a murder being cleared by an arrest as P(cleared)=0.624. For a randomly selected murder, find P(**Multiplication Rule**In this section we present the basic multiplication rule, which is used to find , the probability that event Aoccurs in a first trial and event Boccurs in a second trial.