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limit of probability , first n tosses of unbiased coin

Source: LIMIT 2020 Cat 2 Objective P19

June 9, 2020
probabilitylimticalculus

Problem Statement

Consider an unbiased coin which is tossed infinitely many times. Let AnA_n be the event that no two successive heads occur in the first nn tosses of this experiment. Then which of the following is incorrect : (A) lim⁡n→∞P(An)=0\lim_{n \to \infty} P(A_n)=0
(B) lim⁡n→∞3nP(An)=0\lim_{n \to \infty}3^n P(A_n)=0
(C) 2nP(An)+2n+1P(An+1)=2n+2P(An+22^nP(A_n) +2^{n+1}P(A_{n+1})=2^{n+2}P(A_{n+2}
(D) lim⁡n→∞P(An)P(An+1)\lim_{n \to \infty} \frac{P(A_n)}{P(A_{n+1})} is lesser than 1.21.2
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