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 $x^2+px+q$ are real

• A. $\frac{31}{50}$
• B. $\frac{29}{100}$
• C. $\frac{38}{100}$
• D. $\frac{34}{100}$

Answer 1

The correct option is A $\frac{31}{50}$

Need to be solved case by case
for real roots
$p^2-4q\ge 0$
$p^2\ge 4q$

p	q	Total possible combinations
1	-	0
2	1	1
3	1,2	2
4	1,2,3,4	4
5	1,2,3,4,5,6	6
6	1,2,3,4,5,6,7,8,9	9
7	1,2,3,4,5,6,7,8,9,10	10
8	1,2,3,4,5,6,7,8,9,10	10
9	1,2,3,4,5,6,7,8,9,10	10
10	1,2,3,4,5,6,7,8,9,10	10
	Total	62

No. of ways of choosing $p$ and $q=10.10=$ 100
So $P=$ $\frac{62}{100}$