Question

1.) Each individual in a group of n students is asked to pick an integer at random between 1 and 10 (inclusive).

a.) What is the smallest value of n that assures at least a 50% chance that at least two students select the same number?

b.) What is the smallest value of n that assures at least a 75% chance?

I got P = 1-(10/10)*(9/10)*....*(11-n)/10 but the answers I get don't match up to the answers from the book. If someone could please explain, that would be great!

Answer 1

P( at least two students select the same number )=1-P(all n select different number)

=1-¹⁰P_n/10ⁿ >=0.5

¹⁰P_n/10ⁿ <=0.5

from above for n=1 ¹⁰P₁/10¹   =1

for n=2 ;¹⁰P₂/10² =0.9

for n=3 ;¹⁰P₃/10³ =0.72

for n=4 ;¹⁰P₄/10⁴ =0.504

for n=5 ;¹⁰P₅/10⁵ =0.3024

hence minimum required size n=5

b)

similarly ;

for 1-¹⁰P_n/10ⁿ >=0.75

¹⁰P_n/10ⁿ <=0.25

for n=6 ;¹⁰P₅/10⁵ =0.1512

hence minimum required size n=6