Let f ( n , K ) be the probability that n coins are tossed then K heads are obtained ( K ≤ n ) . Match the following list-I with list-II List - I List - II (P) f ( n , K ) when no 2 heads are consecut

Question

Let f ( n , K ) be the probability that n coins are tossed then K heads are obtained ( K ≤ n ) . Match the following list-I with list-II List - I List - II (P) f ( n , K ) when no 2 heads are consecutive ( n ≥ 2 K − 1) (I) 2 n 2 n − n + K + 1 C K ​ ​ (Q) f ( n , K ) when K heads occur together (II) 2 n 2 n − n − K + 1 C 1 ​ ​ (R) f ( n , K ) when all K heads do not occur together (III) 2 n n − K + 1 C K ​ ​ (S) f ( n , K ) when 2 heads are consecutive (IV) 2 n n − K + 1 C 1 ​ ​ (V) 2 n n − K C 1 ​ ​

TestHub · Accepted Answer

(P) Let ( n − K ) tail occur at any n − K places blank spaces between ( n − K ) tails can be occupied by K heads. Total prob. n − K + 1 C K ​ ( 2 1 ​ ) K ⋅ ( 2 1 ​ ) n − K = n − K + 1 ​ C K ​ (Q) treating the K heads as one unit Prob. = 2 n n − k + 1 C 1 ​ ​ R and S follows from P and Q .

List - I	List - II
(P) $f (n, K)$ when no 2 heads are consecutive ( $n \geq 2 K -$ 1)	(I) $\frac{2 ^{n} - ^{n + K + 1} C _{K}}{2 ^{n}}$
(Q) $f (n, K)$ when K heads occur together	(II) $\frac{2 ^{n} - ^{n - K + 1} C _{1}}{2 ^{n}}$
(R) $f (n, K)$ when all K heads do not occur together	(III) $\frac{^{n - K + 1} C _{K}}{2 ^{n}}$
(S) $f (n, K)$ when 2 heads are consecutive	(IV) $\frac{^{n - K + 1} C _{1}}{2 ^{n}}$
	(V) $\frac{^{n - K} C _{1}}{2 ^{n}}$

Mathematics - Calculus Mixed Question with Solution | TestHub

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