Curriculum Geometry B/ Algebra II
UNIT 6: APPLICATIONS OF PROBABILITY
UNDERSTAND INDEPENDENCE AND CONDITIONAL PROBABILITY AND USE THEM TO INTERPRET DATA |
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MGSE9-12.S.CP.1 Describe categories of events as subsets of a sample space using unions, intersections, or complements of other events (or, and, not).
MGSE9-12.S.CP.2 Understand that if two events A and B are independent, the probability of A and B occurring together is the product of their probabilities, and that if the probability of two events A and B occurring together is the product of their probabilities, the two events are independent.
MGSE9-12.S.CP.3 Understand the conditional probability of A given B as P (A and B)/P(B). Interpret independence of A and B in terms of conditional probability; that is, the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.
MGSE9-12.S.CP.4 Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, use collected data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results.
MGSE9-12.S.CP.5 Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.
MGSE9-12.S.CP.2 Understand that if two events A and B are independent, the probability of A and B occurring together is the product of their probabilities, and that if the probability of two events A and B occurring together is the product of their probabilities, the two events are independent.
MGSE9-12.S.CP.3 Understand the conditional probability of A given B as P (A and B)/P(B). Interpret independence of A and B in terms of conditional probability; that is, the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.
MGSE9-12.S.CP.4 Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, use collected data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results.
MGSE9-12.S.CP.5 Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.
KEY IDEAS
- In probability, a sample space is the set of all possible outcomes. Any subset from the sample space is an event.
- If the outcome of one event does not change the possible outcomes of the other event, the events are independent. If the outcome of one event does change the possible outcomes of the other event, the events are dependent.
- The intersection of two or more events is all of the outcomes shared by both events. The intersection is denoted with the word “and” or with the ⋂ symbol. For example, the intersection of A and B is shown as A ⋂ B.
- The union of two or more events is all of the outcomes for either event. The union is denoted with the word “or” or with the ⋃ symbol. For example, the union of A and B is shown as A ⋃ B. The probability of the union of two events that have no outcomes in common is the sum of each individual probability.
- The complement of an event is the set of outcomes in the same sample space that are not included in the outcomes of the event. The complement is denoted with the word “not” or with the ' symbol. For example, the complement of A is shown as A'. The set of outcomes and its complement make up the entire sample space.
- Conditional probabilities are found when one event has already occurred and a second event is being analyzed. Conditional probability is denoted P(A | B) and is read as “the probability of A given B.” P(A | B) = P(A ⋂ B)/ P(B).
- Two events—A and B—are independent if the probability of the intersection is the same as the product of each individual probability. That is, P(A ⋂ B) = P(A) · P(B).
- If two events are independent, then P(A | B) = P(A) and P(B | A) = P(B).
- Two-way frequency tables summarize data in two categories. These tables can be used to show whether the two events are independent and to approximate conditional probabilities.
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USE THE RULES OF PROBABILITY TO COMPUTE PROBABILITIES
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MGSE9-12.S.CP.6 Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in context.
MGSE9-12.S.CP.7 Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answers in context.
MGSE9-12.S.CP.7 Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answers in context.
KEY IDEAS
- Two events are mutually exclusive if the events cannot occur at the same time.
- When two events A and B are mutually exclusive, the probability that event A or event B will occur is the sum of the probabilities of each event: P(A or B) = P(A) + P(B).
- When two events A and B are not mutually exclusive, the probability that event A or B will occur is the sum of the probability of each event minus the intersection of the two events. That is, P(A or B) = P(A) + P(B) – P(A and B).
- You can find the conditional probability, P(A | B), by finding the fraction of B’s outcomes that also belong to A.
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