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Algebra 2 Section 11.1 Notes Permutations and Combinations The Fundamental Counting Principle describes the method of using multiplication to count. A permutation is an arrangement of items in a particular order. Example to choose 3 items in order, using the fundamental counting principle, there are 3×2×1 = 6 permutations Using factorial notation, you can write 3×2×1 as 3!, read “three factorial”. For any positive integer n, n factorial is n! = n(n-1)(n-2)×…×3×2×1. 0! = 1 The number of permutations of n items of a set arranged r items at a time is: n Pr n! for 0 ≤ r ≤ n (n r )! A selection in which order does not matter is a combination. The number of combinations of n items of a set arranged r items at a time is: n C r Algebra 2 Section 11.2 Notes n! for 0 ≤ r ≤ n r!(n r )! Probability Number of times the event occurs Experimental probability of an event: P(event) = number of trials Sometimes you can’t run actual trials so you can estimate the experimental probability by using a simulation. A simulation is a model of the event. The set of all possible outcomes to an experiment or activity is a sample space. When each outcome in a sample space has the same chance of occurring, the outcomes are equally likely outcomes. If a sample space has n equally likely outcomes and an event A occurs in m of these outcomes, then the theoretical probability of event A is P( A) m . n Algebra 2 Section 11.3 Notes Probability of Multiple Events When the occurrence of one event affects how a second event can occur, the events are dependent events. Otherwise, the events are independent events. For the Probability of A and B, if A and B are independent events, then P(A and B) = P(A)×P(B). Two events that cannot happen at the same time are mutually exclusive events. If A and B are mutually exclusive events, then P(A and B) = 0 For the Probability of A or B, P(A or B) = P(A) + P(B) – P(A and B). If A and B are mutually exclusive events, then P(A or B) = P(A) + P(B) A probability distribution is a function that gives the probability of each outcome in a sample space. You can use a table or graph to represent a probability distribution. A probability distribution that is equal for each event in the sample space is a uniform distribution. Algebra 2 Section 11.4 Notes Conditional Probability The probability that an event, B, will occur given that another event, A, has already occurred is called a conditional probability. Conditional probability exists when 2 events are dependent. You write the conditional probability of event B, given that event A occurs, as P(B|A). You read P(B|A) as “the probability of event B, given event A.” A contingency table is a frequency table that contains data from 2 different categories. Contingency tables and tree diagrams can help you find conditional probabilities. Conditional Probability: for any 2 events A and B with P(A) ≠ 0, P( B |A) P( AandB) . P( A) Algebra 2 Section 11.5 Notes Probability Models You can use probability models to analyze situations and make fair decisions. You can use a probability model to assign probabilities to outcomes of a chance process, which allows you to use probability and simulations to make predictions about real-life situations. Algebra 2 Section 11.6 Notes Analyzing Data Statistics is the study, analysis, and interpretation of data. Measures of central tendency, mean (average), median (# in the middle of the ordered data) and mode (# that occurs the most often) are the most common measures of central tendency. A bimodal data sets has 2 modes. If a data set has more than 2 modes, then the modes are probably not statistically useful. An outlier is a value that is substantially different (usually much larger or much smaller) from the rest of the data. The range of a set of data is the difference between the greatest and least values. If you order data from least to greatest value, the median divides the data into 2 parts. The median of each part divides the data further and you have 4 parts in all. The values separating the 4 parts are quartiles. The interquartile range is the difference between the third and first quartiles. A box-and-whisker plot is a way to display data: Minimum Q1 Median Q3 Maximum A percentile is a number from 0 to 100 that you can associate with a value x from a data set. Algebra 2 Section 11.7 Notes Standard Deviation Standard deviation is a measure of how far the numbers in a data set deviate from the mean. A measure of variation (such as range and interquartile range) describes how the data in a data set are spread out. Variance and standard deviation are measures showing how much data values deviate from the mean. The Greek letter σ (sigma) represents standard deviation. σ2 (sigma squared) is the variance. Variance: 2 (x x) 2 n Algebra 2 Section 11.8 Notes (x x) Standard Deviation: 2 n Samples and Surveys A population is all the members of a set. A sample is part of the population. Sampling Types and Methods: Convenience sample, select members of the population who are conveniently and readily available. Self-selected sample, select only members who volunteer for the sample. Systematic sample, order the population in some way, and then select from it at regular intervals. Random sample, all members of the population are equally likely to be chosen. Study methods: Observational study, you measure or observe members of a sample in such a way that they are not affected by the study. Controlled experiment, you divide the sample into 2 groups. You impose a treatment on one group but not on the other “control” group. Then you compare the effect on the treated group to the control group. Survey, you ask every member of the sample a set of questions. Algebra 2 Section 11.9 Notes Binomial Distributions A binomial experiment has these important features: there are a fixed number of trials; each trial has 2 possible outcomes; the trials are independent; the probability of each outcome is constant throughout the trials. Algebra 2 Section 11.10 Notes Normal Distributions A discrete probability distribution has a finite number of possible events or values. The events for a continuous probability distribution can be any value in an interval of real numbers. A normal distribution has data that vary randomly from the mean. The graph of a normal distribution is a normal curve. A normal distribution has a symmetric bell shape centered on the mean. The margin of error helps you find the interval in which the mean of the population is likely to be. The margin of error is based on the sample and the confidence level desired. A 95% confidence level means that the probability is 95% that the true population mean is within a range of values called a confidence interval. It also means that when you select many different large samples from the same population, 95% of the confidence intervals will actually contain the population mean. The z-score is an important measure for normally distributed data. It indicates the number of standard deviations a value lies above or below the mean of a population. The formula for finding the z-score of a data point of a population is z x , where x is a data point, μ is the mean of the population, and σ is the standard deviation of the population.