A prime number less than 100 is selected randomly. what is the probability that its square has units digit $7$?

Any number ends with 0, 1, 2, 3, 4, 5, 6, 7, 8 or 9

Given any number. The square of this number is the last digit of square of the original numbers units digit.

For example

23*23 ends with 9 (3*3=9)

149*149 ends with 1 (9*9=81)

2564*2564 ends with 6 (4*4=16)

and so on

so all the possible unit digits of a square number are {0, 1, 4, 5, 6, 9}

because:

0*0 = 0; 1*1=1; 2*2=4; 3*3=9; 4*4=16; 5*5=25, 6*6=36; 7*7=49; 8*8=64, 9*9=81

Thus, the probability that the square of a number selected from any set of numbers being 7, is 0.

Answer: 0