Binomial Distbns for students

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Unformatted text preview: © 2010 Radha Bose FSU Department of Statistics Binomial Distributions — 1 ICEBREAKER EXAMPLE Let us suppose that you are planning to have three children. Would you like to have three boys? Or two girls and then a boy? Think of all the possible gender arrangements that you could be fortunate enough to have, and fill in the table below. We will assume that the probability of each child being a girl is 0.6. Gender Arrangement Probability (work) Probability (calculated) B B B 0.4 x 0.4 x 0.4 0.064 B B G 0.4 X 0.4 X 0.6 0.096 B G B 0.4 X 0.6 X 0.4 0.096 G B B 0.6 X 0.4 X 0.4 0.096 B G G 0.4 X 0.6 X 0.6 0.144 G B G 0.6 X 0.4 X 0.6 0.144 G G B 0.6 X 0.6 X 0.4 0.144 G G G 0.6 X 0.6 X 0.6 0.216 Total 1 Now let X = the number of girls you are blessed with . Construct a probability distribution table for X. X Probability 0 1 x 0.064 = 0.064 1 3 x 0.096 = 0.288 2 3 x 0.144 = 0.432 3 1 x 0.216 = 0.216 Total 1 And finally, plot a probability histogram for X by shading in squares in the grid below. Fill in the values of X in the second row from the bottom, and then for each value, shade in a column where the height of the column represents the probability of that value of X. © 2010 Radha Bose FSU Department of Statistics Binomial Distributions — 2 27 18 14 4 Probability: 1 square = 0.016 probability 0 1 2 3 Values of X Notice that the distribution is L E F T-skewed. © 2010 Radha Bose FSU Department of Statistics Binomial Distributions — 3 Bernoulli Trials Bernoulli trials are trials that satisfy the following three conditions: ~ there are only two possible outcomes at each trial (same two for all trials), called "success" and "failure" ~ the probabilities of success and failure are the same for all trials ~ the trials are independent THE BINOMIAL DISTRIBUTIONS ( , ) Bin n π If X = the number of successes that occur in a fixed number of Bernoulli trials , then the appropriate probability model for X is the Binomial model . ~ fixed number of trials ( ) n 4 Binomial ~ only two possible outcomes at each trial (hence the name binomial ) Conditions ~ probability of success ( ) π is the same at all trials, hence probability of failure ( ) 1 π- is also the same at all trials ~ trials are independent ~ model characterized by n and π ~ X is discrete with sample space S = {0, 1, 2, …, n} ~ probabilities obtained by using binomial probability formula ( ) ( ) 1 n r r n r P X r C π π- = = ⋅ ⋅- ~ probability histograms are bell-shaped and exactly symmetric when 0.5 π = , right-skewed when 0.5 π < and left-skewed when 0.5 π > , skewness increases as π moves away from 0.5 ~ tail probabilities decrease as n gets larger (tail area shrinks) IDENTIFYING BINOMIAL SETTINGS Identify which of the settings described below are Binomial. If a setting is not binomial, state why not by choosing one or more of answer choices A, B, C or D. Answers on last page....
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