Probability
Independent Events
Grade 12
Question:
<p>Whenever horses \(a\), \(b\), \(c\) race together, their respective probabilities of winning the race are 0.3, 0.5, and 0.2, respectively. If they race three times, the probability that the same horse wins all the three races, and the probability that \(a\), \(b\), \(c\) each wins one race are, respectively</p>
<p>8/50, 9/50</p>
<p>16/100, 3/100</p>
<p>12/50, 15/50</p>
<p>10/50, 8/50</p>
Step-by-Step Solution
Key Concept: For independent events across multiple trials, multiply probabilities. The same horse winning all 3 races means one specific horse wins each time; each horse winning once means considering all permutations of the outcomes (a,b,c) across 3 races.
<p><strong>Step 1: Probability that same horse wins all three races</strong></p><p>Since races are independent, multiply probabilities for each case:</p><p>• P(a wins all 3) = (0.3)³ = 0.027</p><p>• P(b wins all 3) = (0.5)³ = 0.125</p><p>• P(c wins all 3) = (0.2)³ = 0.008</p><p>Total = 0.027 + 0.125 + 0.008 = <strong>0.160</strong></p><p><strong>Step 2: Probability that a, b, c each wins exactly one race</strong></p><p>One specific order (a wins race 1, b wins race 2, c wins race 3):</p><p>P = 0.3 × 0.5 × 0.2 = 0.03</p><p>Since a, b, c can win in any of 3! = 6 different orders:</p><p>Total = 6 × 0.03 = <strong>0.18</strong></p><p>∴ Answer: A (0.160 and 0.18 or equivalent)</p>
Correct Answer: A