n students filled their forms for a competitive exam. Probability that exactly r students will not appear in the exam is proportional to r. If probability that out of remaining n–r students exactly i students are selected is proportional to i. Prove that the probability of exactly two students finally getting selected is 
Text Solution
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Let E r = event that exactly r students do not appear.
Then P (E r ) = kr
So, P (E 1 ) + P (E 2 ) + ……… + P (E n ) = 1
⇒ k(1 + 2 + ….. + n) = 1 ⇒ k = 
Let A j = event that exactly j students are selected out of n– r
Then P (A j /E r ) = k r j
So, 1.k r + 2k r + ……… + (n – r)k r = 1
⇒ k r = 
Let B = event that exactly two students are selected.
Then P(B)=P(E n – 2 ) P(A 2 /(E n – 2 )) + P(E n – 3 ) P(A 2 /(E n – 3 ))+…….+P(E 1 ) P(A 2 /E 1 )+P(E 0 )P (A 2 /E 0 )
= k(n – 2).2k n – 2 + k(n – 3).2k n – 3 + …… + k.1.2k 1
= 4k 
= 4K 
= 
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