Question

Please show all steps for credit. Thank you. A function H(x) hashes an input x to an integer uniformly distributed between 1 and 10. If there are 100 random inputs, what is the probability that at least two have the same output?

Answer 1

UNIFORM DISTRIBUTION
PDF of uniform distribution f(x) = 1 / ( b - a ) for a < x < b
b = maximum Value
a = minimum Value
f(x) = 1/(b-a) = 1 / (10-1) = 1 / 9 = 0.1111
I.
mean = a + b / 2
=(1+10)/2
=5.5
II.
standard deviation = sqrt ( ( b - a ) ^ 2 / 12 )
=sqrt(10-1)^2 / 12
=2.5981

the probability that at least two have the same output
P(X < 2) = (2-1) * f(x)
= 1*0.1111
= 0.1111
p(X>=2) =1-p(X<2)
p(X>=2) = 1-0.1111
p(X>=2)= 0.8889