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 $x^{2} + p x + q = 0$ are real.___

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Solution

If roots of $x^{2} + p x + q = 0$ are real,
then $p^{2} - 4 q \geq 0$ .....................(1)
Both p,q, belong to set $S = 1, 2, 3 \dots 10$ 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
$= \frac{62}{100} = 0.62$

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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 x2+px+q=0 are real.___

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