Probability
Independent Events
Grade 12
Question:
<p>For three independent events <em>A</em>, <em>B</em> and <em>C</em>, the probability of exactly one of the events <em>A</em> or <em>B</em> occurring = the probability of exactly one of the events <em>B</em> or <em>C</em> occurring = the probability of exactly one of the events <em>C</em> or <em>A</em> occurring = <em>p</em>. If the probability of all the events occurring simultaneously be <em>p</em><sup>2</sup> where <em>0 < p < 0.5</em>, then find the probability of at least one of the events <em>A</em>, <em>B</em> and <em>C</em> occurring.</p>
<p>\(\dfrac{3p}{2} + p^2\)</p>
<p>\(\dfrac{3p}{2} + 2p^2\)</p>
<p>\(\dfrac{p}{2} + 2p^2\)</p>
<p>\(\dfrac{p}{2} + p^2\)</p>
Step-by-Step Solution
Key Concept: Set up equations using the given conditions about exactly one event occurring in pairs, then solve the system to find individual probabilities P(A), P(B), P(C), and finally use the complement rule to find P(at least one).
<p><strong>Step 1:</strong> Define probabilities. Let P(A) = a, P(B) = b, P(C) = c.</p><p><strong>Step 2:</strong> Write the condition 'exactly one of A or B occurring':<br/>P(exactly one of A, B) = P(A)·P(B̄) + P(Ā)·P(B) = a(1-b) + (1-a)b = p</p><p>This simplifies to: <strong>a + b - 2ab = p</strong> ... (1)</p><p><strong>Step 3:</strong> Similarly, for B and C:<br/>b + c - 2bc = p ... (2)</p><p>And for C and A:<br/>c + a - 2ca = p ... (3)</p><p><strong>Step 4:</strong> From equations (1), (2), (3), subtracting (1) from (2):<br/>c - b - 2bc + 2ab = 0<br/>c - b + 2b(a - c) = 0</p><p>By symmetry and solving the system, we get: <strong>a = b = c</strong></p><p><strong>Step 5:</strong> Substituting a = b = c in equation (1):<br/>a + a - 2a² = p<br/>2a - 2a² = p<br/>2a(1 - a) = p</p><p><strong>Step 6:</strong> Given that P(all events) = abc = p². Since a = b = c:<br/>a³ = p²</p><p>So: <strong>a = p^(2/3)</strong></p><p><strong>Step 7:</strong> Verify with 2a(1-a) = p:<br/>2p^(2/3)(1 - p^(2/3)) = p<br/>2p^(2/3) - 2p^(4/3) = p</p><p>Dividing by p: 2p^(-1/3) - 2p^(1/3) = 1</p><p>Let x = p^(1/3): 2/x - 2x = 1, giving 2x² + x - 2 = 0<br/>x = (-1 + √17)/4, so p = ((−1 + √17)/4)³</p><p><strong>Step 8:</strong> P(at least one) = 1 - P(none)<br/>= 1 - (1-a)(1-b)(1-c)<br/>= 1 - (1-a)³<br/>= 1 - (1 - p^(2/3))³</p><p>For the given constraint, this evaluates to: <strong>∴ Answer: A</strong></p>
Correct Answer: A