Probability
Classical Probability
Grade None
Question:
<p>Balls are drawn one-by-one without replacement from a box containing 2 black, 4 white and 3 red balls till all the balls are drawn. Find the probability that the balls drawn are in the order 2 black, 4 white and 3 red.</p>
<p>\(\dfrac{1}{630}\)</p>
<p>\(\dfrac{1}{1260}\)</p>
<p>\(\dfrac{1}{420}\)</p>
<p>\(\dfrac{1}{2520}\)</p>
Step-by-Step Solution
Key Concept: When drawing all balls without replacement in a specific order, the probability equals the product of conditional probabilities at each step, or equivalently: (favorable outcomes)/(total arrangements) = (2!×4!×3!)/(9!), since we need balls of the same color to be grouped in order.
<p><strong>Step 1:</strong> Total ways to arrange 2 black, 4 white, and 3 red balls in 9 positions:</p><p>Since balls of the same color are identical, total arrangements = 9!/(2!×4!×3!)</p><p><strong>Step 2:</strong> Favorable outcomes where order is 2 black, then 4 white, then 3 red:</p><p>There is exactly 1 such arrangement (all blacks first, all whites second, all reds third).</p><p><strong>Step 3:</strong> Calculate probability:</p><p>P = (Favorable outcomes)/(Total arrangements) = 1/[9!/(2!×4!×3!)] = (2!×4!×3!)/9!</p><p><strong>Step 4:</strong> Simplify:</p><p>P = (2×24×6)/(362880) = 288/362880 = 1/1260</p><p>∴ Answer: B (1/1260)</p>
Correct Answer: B