Question

If p and q are chosen randomly from the set {1,2,3,4,5,6,7,8,9,10} with replacement, determine the probability that the roots of the equation $x^2+px+q=0$ are real.___

Answer 1

If roots of$x^2+px+q=0$ are real,
then$p^2-4q\ge 0$ .....................(1)
Both p,q, belong to set $S=1,2,3\cdots 10$ when
p=1, no value of q from S will satisfy (1)
p = 2 q=1 will satisfy
p = 3 q=1,2
p = 4 q=1,2,3,4
p = 5 q=1,2,3,4,5,6
p = 6 q=1,2,3,4,5,6,7,8,9,\)
For p = 7,8,9,10 all the ten values of q will satisfy.
Sum of these selections is
$1+2+4+6+9+10+10+10+10=62$
But the total number of selections of p and q without any order is $10\times 10=100$
Hence the required probability is
$=\frac{62}{100}=0.62$