# From choose to many ways how ie formula ial

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! from choose to many ways how i.e. formula ial combinator the need Here ns) permutatio and ons (combinati techniques counting Recall 3 2 = - = = = = - - = - = C x n n n n n x n x n C n x n x 8

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9 Binomial distribution… n x p p x n x n x P x X P p n x x n x , , 1 , 0 for ) 1 ( )! ( ! ! ) ( ) ( is success of y probabilit & trials with experiment binomial a in successes of y Probabilit ) ( = - - = = - l Members of this family of distributions are characterized by n & p l Different binomial rv’s when have different n & p combinations l But given n & p it is irrelevant if we’re considering a coin toss, family composition, etc (see slide 3)
10 Binomial example l A multiple choice exam has 10 questions each of which has 5 choices of which only one is correct l If a student uses guesswork to answer the quiz, what is: l P ( X = 5)? l P ( X ≤ 1)? l P ( X > 1)? l See Keller Ex 7.9/7.10, pp. 246-7 for probability of failing 3758 . 1 ) 1 ( 1 ) 1 ( 3758 . 2684 . 1047 . ) 8 )(. 2 (. ! 9 ! 1 ! 10 ) 8 (. ) 2 (. ! 10 ! 0 ! 10 ) 1 ( ) 0 ( ) 1 ( 0264 . ) 8 (. ) 2 (. ! 5 ! 5 ! 10 ) 8 (. ) 2 (. ) 5 ( rv binomial a is answers correct of number assume to reasonable & 2 . 0 , 10 Here 9 10 0 5 5 5 5 10 5 - = - = = + = + = = + = = = = = = = = X P X P X P X P X P C X P p n

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Binomial example… l In this example l P( X = 5) is a binomial probability l P( X ≤ 1) is a cumulative binomial probability l P( X > 1) is called a survivor probability l Even with counting techniques, these probabilities can be cumbersome to evaluate l So, values of binomial probabilities have also been tabulated l Tables of individual probabilities & for cumulative probabilities 11
Binomial tables l For P( X =5) when n =10, p =0.2 l Need column labelled p =0.2 in block of values for n=10 l Now consider row labelled k =5  0.0264 as before l Similarly P( X ≤ 1) could be verified from table of individual probabilities by summing as before l Or directly from table of cumulative probabilities 12

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14 Binomial tables… Binomial Probability P ( X = k ) p n k 0.05 0.10 0.15 0.20 0.25 0.90 0.95 2 0 1 2 0.9025 0.0950 0.0025 0.8100 0.1800 0.0100 0.7225 0.225 0.0225 0.6400 0.3220 0.0400 0.5625 0.3750 0.0625 0.0100 0.1800 0.8100 0.0025 0.0950 0.9025 3 0 1 2 3 0.8574 0.1354 0.0071 0.0001 0.7290 0.2430 0.0270 0.0010 0.6141 0.3251 0.0574 0.0034 0.5120 0.3840 0.0960 0.0080 0.4219 0.4219 0.1406 0.0156 0.0010 0.0270 0.2430 0.7290 0.0001 0.0071 0.1354 0.8574 10 0 1 2 3 4 5 6 7 8 9 10 0.5987 0.3151 0.0746 0.0105 0.0010 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.3487 0.3874 0.1937 0.0574 0.0112 0.0015 0.0001 0.0000 0.0000 0.0000 0.0000 0.1969 0.3474 0.2759 0.1298 0.0401 0.0085 0.0012 0.0001 0.0000 0.0000 0.0000 0.1074 0.2684 0.3020 0.2013 0.0881 0.0264 0.0055 0.0008 0.0001 0.0000 0.0000 0.0563 0.1877 0.2816 0.2503 0.1460 0.0584 0.0162 0.0031 0.0004 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0015 0.0112 0.0574 0.1937 0.3874 0.3487 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0010 0.0105 0.0746 0.3151 0.5987

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15 Binomial tables… l Tables provide the entire distribution l What does this distribution
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