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  4. A fair coin is tossed n times. Let p n denotes the probability that no two (or more) consecutive heads occur in n tosses. Statement-1: The probabilities p 2, p 3, p 4 are in arithmetic progression. Statement-2: The probabilities p 1, p 2, p 3, ldots ldots ldots, p n are in decreasing order.

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Q. A fair coin is tossed $n$ times. Let $p _{ n }$ denotes the probability that no two (or more) consecutive heads occur in $n$ tosses.
Statement-1: The probabilities $p _2, p _3, p _4$ are in arithmetic progression.
Statement-2: The probabilities $p _1, p _2, p _3, \ldots \ldots \ldots, p _{ n }$ are in decreasing order.

Probability - Part 2

Solution:

$P_n=P_{n-1} P(T)+P_{n-2} P(T) P(H) ; P_n=\frac{P_{n-1}}{2}+\frac{P_{n-2}}{4} ; P_1=1, P_2=\frac{3}{4} ; P_3=\frac{3}{8}+\frac{1}{4}$
$P _3=\frac{5}{8} ; P _4=\frac{ P _3}{2}+\frac{ P _2}{4} ;=\frac{8}{16}$