Alg1231W12T1 - Applications of the binomial distribution

Saturday, 20 October 2018

10:46 PM

 

 

Machine generated alternative text: 25 0.5 Pk 0.4 0.3 0.2 0.1 p 10 5 The binomial distribution B(n,p) for a fixed p (with 0 < p < 1) and n e N is given by 0.5 15 n 20 25 k for k = O, 1, 2, . . , n. Here Pk represents the probability of tossing a coin; whose probability of getting heads is p, and obtaining exactly k heads out of n tosses Cl n heads out of k tosses Cl k tails out of n tosses Cl n tails out of k tosses. If the coin is fair; then p 0.5 O In this special case the probability is So for a fair coin, the probability of getting exactly 4 heads from 6 tosses is precisely 15/64 Note: the Maple notation for the binomial coefficient is binomial (a , b) b

 

Machine generated alternative text: Annie wins 70% of her tennis matches. Assuming each match is an independent random event; we can model Annie's tournaments using a binomial distributiom On a weekend tournament where she plays 6 matches: i) The probability that Annie wins 4 matches is exactly 64827/200000 ii) The probability that she wins fewer than three matches is exactly 7047/100000 Note: the Maple notation for the binomial coefficient is binomial (a,b) b which to two decimal places is approximately 082 O fib, which to two decimal places is approximately 0 070

 

Machine generated alternative text: 0.5 Pk 0.4 0.3 0.2 0.1 5 n 18 P = 0.75 15 20 25 S x 10) 0.06 The GeoGebra Applet above allows you to explore a binomial distribution up to n = 25. Suppose Bob wins 75% of his professional chess matches. He going to play 18 of chess games next week. Determine the following. i) The probability that Bob wins between 10 and 15 games is (to two decimal places) 0.85 Note: here between means includuing 10 and 15 ii) The probability that Bob wins 17 or more games is (to ftvo decimal places) 04 iii) The probability that Bob wins 10 or less games is .06

 

Machine generated alternative text: The two most crucial numbers associated to a probability distribution are: If p Maximum probability Mean (or expected value) Niceness Variance 0.5 and n = 12 then the probability distribution B(12, 0.5) has mean 6 O np(l-p) 2 and variance o 3 In general B(n,p) has mean 2 and variance

 

Machine generated alternative text: A test contains 10 multiple choice questions; each with four possible answers. If a student was to simply guess what the answers are, then we can model the final number of questions this student gets correct as a binomial distribution B(n,p) with n 10 O and p = 0.25 If the pass mark is 5 out of 10 what is the probability that this student will pass the test? It is (to 5 decimal places) 0178126907: What is the probability the student gets at most 2 out of 10? It is (to 5 decimal places) What is the correct plot for the probability distribution in this problem? 0525592804( 0.75 O 095 0.75 0.25 0.75 O 095 e 10 11 1 e 10 11 1 e 10 11 1

 

Machine generated alternative text: Likewise; what is the correct plot for the cumulative distribution function? O 095 10 11 12 O 10 11 12 10 11 12

 

 

Machine generated alternative text: binomial(Lk) pk (1 — end proc evalf( 2 ) evalf( 2 ) binomial(Lk) (1 64827 200000 0.32 7047 0.070 k) end proc

 

 

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