First slide

Introduction to probability

Question

If p and q are chosen randomly from the set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} with replacement, then the probability that the roots of the equation x² + px + q = 0

Moderate

A

are real is 33/50
B

are imaginary is 19/50
C

are real and equal is 3/50
D

are real and distinct is 3/5

Solution

Roots of x² + px + q = 0 will be real if $p^{2} \geq 4 q$ .
The possible selections are as follows:
p q
1 -
2 1
3 1, 2
4 1, 2, 3, 4
5 1, 2, 3, 4, 5, 6
6 1, 2, …….., 9
7 1, 2, …….., 10
8 1, 2, …….., 10
9 1, 2, …….., 10
10 1, 2, …….., 10
Total 62
Therefore, number of favourable ways is 62 and total number of ways is 10² = 100. Hence, the required probability is
62/100 = 31/50.
The probability that the roots are imaginary is
1 - 31/50 = 19/50.
Roots are equal when (p, q) $\equiv$ (2, 1), (4, 4), (6,9). The probability that the roots are real and equal is 3/50. Hence, probability that the roots are real and distinct is 3/5.

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App