7. An urn contains 20 slips of paper numbered 1 through 20. An experimenter draw

Question

7. An urn contains 20 slips of paper numbered 1 through 20. An experimenter draws out four slips of paper at random with replacement. What is the probability that (a) Only one slip of paper displays a prime number? (b) At least one slip of paper displays a prime number? (c) If 5000 experimenters were each to independently perform this same experiment of four draws and individually report the number of primes that each obtains to a research organization, what is the total number of primes that the research organization would receive from the 5000 experimenters?

Explanation / Answer

Out of 1 to 20:

There are 8 Prime numbers: 2,3,5,7,11,13,17,19

and rest 12 numbers are not prime

If four slips are drawn

a) P(only one slip displays prime number) =

(ways of selecting 1 prime number from 8 * ways of selecting 3 non prime numbers from 12)/ways of selecting 4 out of 20

= 8C1*12C3 / 20C4

= 8 * (12*11*10/3!) / (20*19*18*17/4!)

=1760/4845 = 0.363

b) P(atleast one slip displays prime number) = 1 - P(no slip displays prime number)

= 1 - 8C0*12C4 / 20C4

= 1 - ( (1*12*11*10*9/4!) / (20*19*18*17/4!)

= 1 - 0.102

= 0.898

C)

Probability of getting a prime number on 1st/2nd/3rd/4th draw = 8/20 = 2/5 = 0.4 (as with replacement)

Total experiments = 5000

Total drwas = 5000* 4 = 20000

Expectation of prime numbers = Total draws * P(prime number)

= 20000*0.4

= 8000

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