Random numbers?

from the numbers 3 to 12 inclusive, two numbers are chosen at random.


a) what is the cardinality of the sample space?


b) list the event in which one number is a factor of the other number and find its cardinality


c) list the event in which the sum of the numbers is 15


d) list the event in which the sum of the numbers is 15 and one number is a factor of the other number

Random numbers?
a) The cardinality of a set is simply the number of elements in the set. An element of this sample space is an ordered pair of numbers from 3 to 12, such as (5,6) or (7,9). There are ten possibilities for each selection, and two selections. Since repeats are allowed (unless stated otherwise), the number of total posibilities we have is 10x10=100. Thus, the cardinality of the sample space is 100.





b) Since we are only going up to 12, we know that only numbers going up to 6 (half of 12) can be a factor of a number greater than itself. Then we add in all the "doubles" (such as (6,6)), since every number is a factor of itself. So our possibilities are (3,3), (3,6), (3,9), (3,12), (4,4), (4,8), (4,12), (5,5), (5,10), (6,6), (6,12), (7,7), (8,8), (8,4), (9,9), (9,3), (10, 10), (10,5), (11,11), (12,12), (12,6), (12,3). So we count them up and that's our cardinality. So the answer is 22.





c) Based on the first selection, there is exactly one possibility for the second number that gives a sum of the two as 15. So the elements with sum of 15 are (3,12), (4,11), (5,10), (6,9), (7,8), (8,7), (9,6), (10,5), (11,4), (12,3).





d) Now we are looking for the elements which meet the criteria of both parts b and c, so we are looking for the intersection of the two sets. So this means we take every element that appears on both lists. So the answer is (3,12),(5,10), (10,5), (12,3).

curse of the golden flower