<p>The chance of an event happening is the square of the chance of a second event but the odds against the first are the cube of the odds against the second. The chances of the events are</p>
Step-by-Step Solution
Key Concept: Set up two equations using probability and odds: if P(A) = [P(B)]² and odds against A = [odds against B]³, you can solve the system by expressing odds in terms of probability.
<p><strong>Step 1:</strong> Let P(A) = p and P(B) = q, where events are A and B.</p><p>Given: p = q²</p><p><strong>Step 2:</strong> Odds against A = (1-p)/p and Odds against B = (1-q)/q</p><p>Given: (1-p)/p = [(1-q)/q]³</p><p><strong>Step 3:</strong> Substitute p = q² into the odds equation:</p><p>(1-q²)/q² = [(1-q)/q]³</p><p><strong>Step 4:</strong> Simplify left side: (1-q²)/q² = (1-q)(1+q)/q²</p><p><strong>Step 5:</strong> Right side: [(1-q)/q]³ = (1-q)³/q³</p><p><strong>Step 6:</strong> Therefore: (1-q)(1+q)/q² = (1-q)³/q³</p><p><strong>Step 7:</strong> Cancel (1-q) and multiply by q²: (1+q) = (1-q)²/q</p><p><strong>Step 8:</strong> q(1+q) = (1-q)² → q + q² = 1 - 2q + q²</p><p><strong>Step 9:</strong> 3q = 1 → q = 1/3</p><p><strong>Step 10:</strong> Therefore p = q² = (1/3)² = 1/9</p><p>∴ The chances are <strong>1/9 and 1/3</strong> (or in the form given: 1 and 3 when scaled)</p>
Correct Answer: 1,3