30 Wyner Statistics Fall 2013

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1 30 Wyner Statistics Fall 2013 CHAPTER FIVE: DISCRETE PROBABILITY DISTRIBUTIONS Summary, Terms, and Objectives A probability distribution shows the likelihood of each possible outcome. This chapter deals with discrete probability distributions, in which there are a set number of possible outcomes in any given range (e.g., no cars, 1 car, 2 cars), as opposed to continuous probability distributions, in which there are infinitely many possible outcomes which may be sorted into discrete categories (e.g., 0 to 10 cm, 10 to 20 cm, 20 to 30 cm). A particularly important discrete probability distribution is the binomial probability distribution, in which a trial such as predicting a coin flip is attempted n times and P(r) represents the probability of exactly r of those n trials being successful. 5-A Introduction to Probability Distributions Wednesday 11/20 discrete variable continuous variable probability distribution ➊ Classify variables as discrete or continuous. ➋ Give the probability distribution of an event. ➌ Calculate the mean and standard deviation of a discrete probability distribution. 5-B Geometric and Binomial Probabilities Tuesday 11/26 geometric probability distribution binomial probability distribution binomial experiment ➊ Calculate the probability that the first success will be on the n th trial. ➋ Calculate the probability that the first success will be after the n th trial. ➌ Calculate the probability of getting exactly r successes in a binomial experiment. ➍ Calculate the probability of getting at most or at least r successes in a binomial experiment. ➎ Calculate binomial probabilities with the calculator. 5-C Binomial Distributions Wednesday 12/4 ➊ Make a histogram for a binomial probability distribution. ➋ Calculate the mean and standard deviation for number of successes in a binomial distribution. Review Friday 12/6 Test Tuesday 12/17

2 31 Wyner Statistics Fall A Introduction to Probability Distributions Within any given range, a DISCRETE Variable has a limited number of possibilities and a CONTINUOUS Variable has infinite possibilities. ➊ Classify variables as discrete or continuous. 1. Data that are counted are discrete. 2. Most data that are measured or that need to be rounded are continuous. ➊ Classify the following as discrete or continuous. a) Shoe size (9, 9.5, 10, etc.) does not need to be rounded so it is discrete. b) Foot size (25.6 cm, cm, etc.) is measured and must be rounded so it is continuous. c) Age (16.83, 17.82, etc.) must be rounded so it is continuous. d) Age rounded down (16, 17, etc.) is already rounded so it is discrete. e) Number of people in a classroom would be counted so it is discrete. f) Number of people in a country would be too hard to count and would likely be rounded so it could be considered continuous. A PROBABILITY DISTRIBUTION shows all the possible outcomes of an event and how likely each one is. The sum of the probabilities in a probability distribution is 1 (that is, 100%) because it includes all possibilities. ➋ Give the probability distribution of an event. 1. List each possible outcome. 2. State the probability of each. ➋ Show the probability distribution for a coin flip. 1. heads, tails 2. 50%, 50%

3 32 Wyner Statistics Fall 2013 The mean of a probability distribution is µ = xp(x). This is also the expected value. The standard deviation is of a probability distribution is ß = (P(x)(x µ) 2 ). These formulas are the same as those in 3-C, except using P(x) instead of f. ➌ Calculate the mean and standard deviation of a discrete probability distribution. 1. List each possible outcome x and its probability P(x). 2. Calculate xp(x) for each outcome. 3. To get the mean, find the sum of the xp(x) s. 4. Subtract µ from each possible outcome. 5. Square each result in step Multiply each square in step 5 by that outcome s probability P(x). 7. To get the variance, add the products in step To get the standard deviation, take the square root of the variance in step 7. ➌ At Harris High, 14% of the seniors take 4 classes, 58% take 5, 20% take 6, and 8% take 7. x P(x) xp(x) x µ (x µ) 2 P(x) (x µ) µ = 5.22 ß 2 = (P(x)(x µ) 2 ) = ß = = 0.782

4 33 Wyner Statistics Fall B Geometric and Binomial Probabilities In the probability distributions below, n represents the number of trials, r represents the number of successes, p represents the probability of success on each trial (and must be the same for each trial), and q represents the probability of failure on each trial. p and q are complements. The GEOMETRIC Probability Distribution, P(n) = q n 1 p, gives the probability that the first success will be the n th trial. Similarly, P(n) = q n gives the probability that the first success will be after the n th trial. ➊ Calculate the probability that the first success will be on the n th trial. 1. Identify n, the number of trials. 2. Identify p, the probability of success on each individual trial. 3. Identify q, the probability of failure on each individual trial. 4. Calculate P(n) = q n 1 p. ➊ Nadene is predicting rolls on 8-sided dice. Find the probability that her first successful prediction will be her fourth roll. 1. There are n = 4 rolls. 2. The probability of a getting an 8 is p = 1 / 8 on each roll. 3. The probability of not getting an 8 is q = 7 / 8 on each roll. 4. P(4) = ( 7 / 8 ) 3 ( 1 / 8 ) = 243 / 4096 ➋ Calculate the probability that the first success will be after the n th trial. 1. Identify n and q (see steps 1 and 3, above). 2. Calculate P(n) = q n. ➋ Nadene is predicting rolls on 8-sided dice. Find the probability that her first successful prediction will be after her fourth roll. 1. n = 4, q = 7 / 8 2. P(4) = ( 7 / 8 ) 4 = 2401 / 4096 The BINOMIAL Probability Distribution, P(n) = ( n r )p r q n-r, gives the probability that exactly r out of n trials will be successes. This setup is called a BINOMIAL EXPERIMENT. ➌ Calculate the probability of getting exactly r successes in a binomial experiment. 1. Identify n, the number of trials. 2. Identify r, the number of successes. 3. Identify p, the probability of success on each individual trial. 4. Identify q, the probability of failure on each individual trial. 5. Calculate P(n) = ( n r )p r q n-r. ➌ Find the probability that out of 5 6-sided dice exactly 3 will roll There are n = 5 rolls. 2. There are r = 3 6 s. 3. The probability of a getting a 6 is p = 1 / 6 on each roll. 4. The probability of not getting a 6 is q = 5 / 6 on each roll. 5. P(3) = ( 5 3) ( 1 /6) 3 ( 5 /6) 2 = 10( 1 /216) ( 25 /36) = 250 /7776

5 34 Wyner Statistics Fall 2013 ➍ Calculate the probability of getting at most or at least r successes in a binomial experiment. 1. Identify n, p, and q (see steps 1, 3, and 4, above). 2. Identify the range for r. 3. Do step 5, above, for each value of r in the range stated. 4. Add together the results of step 3. ➍ Find the probability that out of 5 6-sided dice at least 3 will roll n = 5, p = 1 / 6, q = 5 / 6 2. At least 3 out of 5 means r = 3, 4, or P(3) = ( 5 3) ( 1 /6) 3 ( 5 /6) 2 = 10( 1 /216) ( 25 /36) = 250 /7776 P(4) = ( 5 4) ( 1 /6) 4 ( 5 /6) 1 = 5( 1 /1296) ( 5 /6) = 25 /7776 P(5) = ( 5 5) ( 1 /6) 5 ( 5 /6) 0 = 1( 1 /7776) (1) = 1 / P(3, 4 or 5) = 250 / / /7776 = 276 /7776 Binomial probabilities can be calculated directly on the calculator. binompdf (n, p, r) is the probability of exactly r successes. binomcdf (n, p, r) is the probability of at most r successes. 1 binomcdf (n, p, r 1) is the probability of at least r successes, which is the complement of at most r 1 successes. ➎ Calculate binomial probabilities with the calculator. 1. Identify n, r, and p. 2. Choose the appropriate function above, depending on whether it is an exactly at most or at least problem. 3. Plug in the variables. ➎ Find the probability that out of 5 6-sided dice at least 3 will roll n = 5, r = 3, p = 1 / binomcdf (n, p, r 1) 3. P(3, 4, or 5) = 1 binomcdf (5, 1 / 6, 2) = 3.55%

6 35 Wyner Statistics Fall C Binomial Distributions A binomial probability distribution can be shown in a histogram, in which each possible value of r has a bar showing its probability. ➊ Make a histogram for a binomial probability distribution. 1. Identify n, p, and q. 2. Calculate the height of each bar (0 through n) by binompdf(n, p, r). 3. Label the x-axis number of successes from 0 to n, and the y-axis P(r), starting at 0%. 4. Graph each bar. 5. Title the graph. ➊ Give the probability distribution for predicting 4 rolls on a 4-sided die. 1. n = 4, p = 1 /4, q = 3 /4 2. P(0) = binompdf(4, 1 /4, 0) =.316 P(1) = binompdf(4, 1 /4, 1) =.422 P(2) = binompdf(4, 1 /4, 2) =.211 P(3) = binompdf(4, 1 /4, 3) =.047 P(4) = binompdf(4, 1 /4, 4) =.003 The mean and expected value of a binomial distribution is µ = np. This is also the most likely result. The standard deviation of a binomial distribution is ß = npq. ➋ Calculate the mean and standard deviation for number of successes in a binomial distribution. 1. Identify n, p, and q. 2. Calculate µ = np. 3. Calculate ß = npq. ➋ Find the mean and standard deviation for number of correct predictions out of 4 rolls on a 4-sided die. 1. n = 4, p = 1 /4, q = 3 /4 2. µ = 4( 1 /4) = 1 ß = 4( 1 /4)( 3 /4) =.866 Probability P(r) 50% 40% 30% 20% 10% 0% Probability Distribution for Number of Correct Predictions on 4 rolls of a 4-Sided Die # of Successful Predictions r

Chapter 8: The Binomial and Geometric Distributions

Chapter 8: The Binomial and Geometric Distributions 8.1 Binomial Distributions 8.2 Geometric Distributions 1 Let me begin with an example My best friends from Kent School had three daughters. What is the