A First Course in Probability- Sheldon Ross 8th Edition Random Variables Example 1b

Question

Three balls are to be randomly selected without replacement from an urn containing
20 balls numbered 1 through 20. If we bet that at least one of the balls that are
drawn has a number as large as or larger than 17, what is the probability that we
win the bet?

Comments

other way it can be done as

3c1*(4/20)*(16/19)*(15/18)+3c2*(4/20)*(3/19)*(16/18)+(4/20)*(3/19)*(2/18)=0.5087

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http://faculty.washington.edu/fscholz/FallMathStat394A_2011/LEC12-21.pdf

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https://math.stackexchange.com/questions/2767202/probability-that-if-three-balls-numbered-from-1-20-are-selected-without-replacem

Answer 1

This question is from hypergeometric distribution basically which has the following features :

a) The populatin size which is also known sample space size is finite .

b) Objects are drawn without replacement i.e. once an object is drawn is not put back..

c) Hence probability of drawing each object varies unlike binomial distribution..

So here we have :

P(at least one ball value >= 17)       =  1 - P(all ball values <= 16)

                                                    =   1 - ( ¹⁶C₃ /  ²⁰C₃ )

                                                    =   1 - [ ( 16 * 15 * 14 ) / (20 * 19 * 18) ]

                                                    =   0.50877

                                                    =   0.509

Hence required probability              =   0.509

Answer 2

Atleast one of the balls larger than 1 = 1 - (None of the balls larger than 1)

                                                                  = 1-((16*15*14)/(20*19*18))

                                                                  =  0.508