<p>An urn contains five balls. Two balls are drawn and are found to be white. Find the probability that all the balls are white.</p>
Step-by-Step Solution
Key Concept: Use Bayes' theorem to find P(all white | 2 white drawn). The probability depends on the prior distribution of white balls in the urn, which requires considering all possible compositions that could result in drawing 2 white balls.
<p><strong>Step 1:</strong> Define events. Let A = 'all 5 balls are white' and B = '2 balls drawn are white'.</p><p><strong>Step 2:</strong> We need P(A|B) = P(B|A)·P(A)/P(B) using Bayes' theorem.</p><p><strong>Step 3:</strong> Without loss of generality, assume uniform prior: possible compositions are 2, 3, 4, or 5 white balls (each with prior probability). The question implies we use the principle of indifference.</p><p><strong>Step 4:</strong> Calculate P(B|A) for each case:</p><ul><li>If 5 white (all white): P(draw 2 white) = C(5,2)/C(5,2) = 1</li><li>If 4 white, 1 black: P(draw 2 white) = C(4,2)/C(5,2) = 6/10 = 3/5</li><li>If 3 white, 2 black: P(draw 2 white) = C(3,2)/C(5,2) = 3/10</li><li>If 2 white, 3 black: P(draw 2 white) = C(2,2)/C(5,2) = 1/10</li></ul><p><strong>Step 5:</strong> Using equal priors P(A)=1/4 for each case:</p><p>P(B) = (1/4)[1 + 3/5 + 3/10 + 1/10] = (1/4)[10/10 + 6/10 + 3/10 + 1/10] = (1/4)·(20/10) = 1/2</p><p><strong>Step 6:</strong> P(A|B) = P(B|A)·P(A)/P(B) = (1)·(1/4)/(1/2) = 1/4 · 2 = <strong>1/2</strong></p>
Correct Answer: 1/2