SOLUTION: Five-digit codes are selected at random from the set {0, 1, 2, ..., 9} with replacement. If the random variable X denotes the number of zeros in randomly chosen codes, then what ar

The first sentence means that repetitions are allowed in the code's digits.


The possible values for the random variable X are {0, 1, 2, 3, 4, 5}.



X= 0  means that there is NO zero among five digits of the code.

      The probability do not have zero in any one fixed/selected of 5 positions is  ;

      The provability do not have zero in ALL        5 positions, is, therefore,  .



X= 1  means that there is exactly one 0 among five digits of the code.

      The probability for it is   = .



X= 2  means that there is exactly two 0 among five digits of the code.

      The probability for it is  .


The pattern is just clear.  The probability for X = k,  where  k= 0, 1, 2, 3, 4, 5,  is

            .   

    Indeed, this sum is equal to   =  = 1.