For any finite set S, there is a probability function on the set of subsets of S

Question

For any finite set S, there is a probability function on the set of subsets of S given by P(A ) = #(A )/#(S ). If the set is huge, we need counting tricks (like trees and combinations) to compute #(A ); this is the focus in the text. But readers should not lose track of the easy fact that if the set is small, one can determine probabilities just be enumeration. This exercise describes a small sample space, and asks questions based on this probability.

Imagine an experiment which produces an ordered triple of natural numbers (x, y, z ) such that x ? y? z and x +y +z = 18. (Note: the natural numbers are 1,2, 3... . Specifically, 0 is not allowed.) Let Sbe the set of such triples, and let P be the probability discussed above.

1) What is the size of the sample space?

2) What is the chance that a selected triple (x, y, z ) has x = 1?

3) What is the chance that a selected triple (x, y, z ) has y = 4?

4) What is the chance that a selected triple (x, y, z ) has y = z ?

Explanation / Answer

a) we have ordered triplets of natural numbers x,y,z such that

x+y+z= 18

the no.of cases of selecting x=9c1

the no. of cases of selecting y= 9c1

the no of cases of secting z=1

x,y,z then can be ordered one after the other making 6 possible combinations

thus

9c1*9c1*6

then we have to subtract the cases in which two of x, y, z are repeating that are when z=2, 4,6,8,10,12, 14, 16

thus the total sample space would be (9c1*9c1*6)-8 = 108-8=100

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