6. A professor hopes to identify students in her class by the last 4 digits of t

Question

6. A professor hopes to identify students in her class by the last 4 digits of their UINs. Assume each of the 10,000 configurations 0000 to 9999 are equally likely (a) Provide an expression for the probability that at least two students in a class of size n (b) The professor has 15 students in her class. What is the chance that at least two have the same last 4 digits!? (c) What is the smallest class size for which the probability that at least two students have the same 4 digits is at least 0.10?

Explanation / Answer

a)

here total number of ways n stufent can have UIN;s =N(each student has 10000 possible configuation)

=(10000)n

number of ways so that all n student have different UIN's =N(selecting n different number from 10000 UIN's) =10000Pn

hence P(at least 2 have same 4 digits)=1-P(none have same digits) =1-10000Pn/(1000)n

b)

from above P(at least 2 have same last 4 digit)= 1-10000P15/(1000)15

=

1-(10000*9999*9998*9997*9996*9995*9994*9993*9992*9991*9990*9989*9988*9987*9986)/(10000)n

=0.01045

c)

from above 1-10000Pn/(1000)n >0.10

solving above with permutation and combination:

smallest class size n=47

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