<p>Match the following:</p><table border='1' cellpadding='4'><tr><th>Column I</th><th>Column II</th></tr><tr><td>(A) A bag contains 4 red, & 6 white two balls are drawn at random. What is the probability that one of them is red and other is white</td><td>(p) 31/32</td></tr><tr><td>(B) A bag contains 6 apples, 4 bananas and 8 mangoes. If three fruits are drawn at random, then what is the probability that all the three are apples</td><td>(q) 8/15</td></tr><tr><td>(C) Five coins are tossed together. What is the probability that at least one tail will appear?</td><td>(r) 75/198</td></tr><tr><td>(D) Team of 5 is to be constituted out of 6 girls and 6 boys then what is probability that team has 3 girls 2 boys</td><td>(s) 5/204</td></tr></table>
Step-by-Step Solution
Key Concept: Match each probability scenario by correctly identifying the sample space (combinations or total outcomes) and favorable outcomes. For drawing problems, use combination formulas C(n,r); for coin tosses, use 2^n total outcomes; for team selection, use the hypergeometric distribution.
<p><strong>Step 1 (A):</strong> Bag: 4 red, 6 white (total 10). Draw 2 balls, one red and one white.</p><p>Favorable outcomes = C(4,1) × C(6,1) = 4 × 6 = 24</p><p>Total outcomes = C(10,2) = 45</p><p>P = 24/45 = 8/15 → <strong>Answer: q</strong></p><p><strong>Step 2 (B):</strong> Bag: 6 apples, 4 bananas, 8 mangoes (total 18). Draw 3 fruits, all apples.</p><p>Favorable outcomes = C(6,3) = 20</p><p>Total outcomes = C(18,3) = 816</p><p>P = 20/816 = 5/204 → <strong>Answer: s</strong></p><p><strong>Step 3 (C):</strong> Five coins tossed. At least one tail appears = 1 - P(all heads).</p><p>P(all heads) = (1/2)^5 = 1/32</p><p>P(at least one tail) = 1 - 1/32 = 31/32 → <strong>Answer: p</strong></p><p><strong>Step 4 (D):</strong> Team of 5 from 6 girls and 6 boys: select 3 girls and 2 boys.</p><p>Favorable outcomes = C(6,3) × C(6,2) = 20 × 15 = 300</p><p>Total outcomes = C(12,5) = 792</p><p>P = 300/792 = 75/198 → <strong>Answer: r</strong></p><p><strong>∴ A-q, B-s, C-p, D-r</strong></p>
Correct Answer: A-q, B-s, C-p, D-r