Probability
Conditional Probability and Independence
Grade 12
Question:
<p>Let \(C_1\) and \(C_2\) be two biased coins such that the probabilities of getting head in a single toss are \(\dfrac{2}{3}\) and \(\dfrac{1}{3}\), respectively. Suppose \(\alpha\) is the number of heads that appear when \(C_1\) is tossed twice, independently, and suppose \(\beta\) is the number of heads that appear when \(C_2\) is tossed twice, independently. Then the probability that the roots of the quadratic polynomial \(x^2 - \alpha x + \beta\) are real and equal, is:</p>
<p>\(\dfrac{40}{81}\)</p>
<p>\(\dfrac{20}{81}\)</p>
<p>\(\dfrac{1}{2}\)</p>
<p>\(\dfrac{1}{4}\)</p>
Step-by-Step Solution
Key Concept: For a quadratic $ax^2 + bx + c$ to have real and equal roots, the discriminant $D = b^2 - 4ac$ must be zero.
<p>Let $\alpha \sim Bin(2, 2/3)$ and $\beta \sim Bin(2, 1/3)$. Condition: $\alpha^2 = 4\beta$.</p><ul><li>$\alpha=0, \beta=0: P = (1/9)(4/9) = 4/81$</li><li>$\alpha=2, \beta=1: P = (4/9)(4/9) = 16/81$</li></ul><p>Total = 20/81.</p>
Correct Answer: B