Probability
Classical Probability
Grade None
Question:
<p>Whenever horses \(A\), \(B\), \(C\) race together, their respective probabilities of winning are \(\dfrac{1}{2}\), \(\dfrac{1}{5}\), and \(\dfrac{1}{4}\). If they race three times, the probability that the same horse wins all three races is</p>
<p>\(\dfrac{21}{200}\)</p>
<p>\(\dfrac{1189}{8000}\)</p>
<p>\(\dfrac{3}{8}\)</p>
<p>\(\dfrac{1}{8}\)</p>
Step-by-Step Solution
Key Concept: P(same horse wins all 3) = P(A wins all 3) + P(B wins all 3) + P(C wins all 3). Races are independent.
<p>$P(A \text{ wins all 3}) = \left(\dfrac{1}{2}\right)^3 = \dfrac{1}{8}$</p><p>$P(B \text{ wins all 3}) = \left(\dfrac{1}{5}\right)^3 = \dfrac{1}{125}$</p><p>$P(C \text{ wins all 3}) = \left(\dfrac{1}{4}\right)^3 = \dfrac{1}{64}$</p><p>LCM(8, 125, 64) = 8000.</p><p>$P = \dfrac{1000 + 64 + 125}{8000} = \dfrac{1189}{8000}$</p>
Correct Answer: B