Probability
Conditional Probability and Events
Grade 12
Question:
<p>The probability that roots of the quadratic equation \(ax^2 + bx + c = 0\) are imaginary, is</p>
<p>(a) \(\frac{103}{216}\)</p>
<p>(b) \(\frac{133}{216}\)</p>
<p>(c) \(\frac{157}{216}\)</p>
<p>(d) \(\frac{173}{216}\)</p>
Step-by-Step Solution
Key Concept: The probability of imaginary roots equals 1 minus the probability of real roots. Real roots occur when the discriminant is non-negative.
<p><strong>Solution:</strong> Let $p_3$ = Probability that roots of $ax^2 + bx + c = 0$ are imaginary.</p><p>$p_3 = 1 - \text{(Probability that roots are real)}$</p><p>$= 1 - (p_1 + p_2)$</p><p>$= 1 - \left(\frac{43}{216} + \frac{5}{216}\right)$</p><p>$= 1 - \frac{48}{216} = \frac{168}{216} + \frac{5}{216} = \frac{173}{216}$</p><p>∴ Answer is (d) $\frac{173}{216}$</p>
Correct Answer: D