Lesson Ten pairs of shoes are kept in a rack

Ten pairs of shoes are kept in a rack

Problem 1

Let's assume that the shoes are numbered by integer numbers from 1 to 20 in a way that 

    the first pair  is (1,2),

    the second pair is (3,4),

    the third pair  is (4,5),

       . . . and so on . . . 

    the tenth pair is (19,20).


The full space of samples is the space of quadruples of different shoes taken from 20 at a time.


    (it is the same as the set of all subsets consisting of 4 elements of the original set of 20 shoes).


So, the total number of elements of this sample space is   =  = 4845.


Very good.


Now I want to figure out, in how many ways I can choose 4 shoes in a way that there in NO pair/pairs among them.


Obviously, for it, I should choose these 4 shoes from DIFFERENT PAIRS.


In this way, I get the number of such quadruples  as   =  = 3360.

The factor  is due to the fact that there is the choice of one of two shoes inside each pair.



So, the probability under the question is   =  =  = 0.3065 = 30.65%.   ANSWER