<p>If odds against solving a question by three students are 2:1, 5:2 and 5:3, respectively, then probability that the question is solved only by one student is</p>
Step-by-Step Solution
Key Concept: Convert odds to probabilities for each student, then find the probability exactly one solves it by taking: P(A solves AND B,C don't) + P(B solves AND A,C don't) + P(C solves AND A,B don't)
<p><strong>Step 1: Convert odds against to probability of solving</strong></p><p>For student A: odds against = 2:1 ⟹ P(A solves) = 1/(2+1) = 1/3, P(A fails) = 2/3</p><p>For student B: odds against = 5:2 ⟹ P(B solves) = 2/(5+2) = 2/7, P(B fails) = 5/7</p><p>For student C: odds against = 5:3 ⟹ P(C solves) = 3/(5+3) = 3/8, P(C fails) = 5/8</p><p><strong>Step 2: Find probability exactly one student solves</strong></p><p>P(exactly one solves) = P(A only) + P(B only) + P(C only)</p><p>= P(A)·P(B')·P(C') + P(A')·P(B)·P(C') + P(A')·P(B')·P(C)</p><p>= (1/3)(5/7)(5/8) + (2/3)(2/7)(5/8) + (2/3)(5/7)(3/8)</p><p>= 25/168 + 20/168 + 30/168</p><p>= 75/168 = 25/56</p><p><strong>∴ Answer: C</strong></p>
Correct Answer: C