Question

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 $p_1,p_2,p_3,\dots \dots \dots ,p_n$ are in decreasing order.

• A. Statement- 1 is true, statement- 2 is true and statement- 2 is correct explanation for statement- 1 .
• B. Statement- 1 is true, statement- 2 is true and statement- 2 is NOT the correct explanation for statement- 1 .
• C. Statement- 1 is true, statement- 2 is false.
• D. Statement- 1 is false, statement- 2 is true.

Answer 1

Answer: (B)

$P_n=P_{n-1}P(T)+P_{n-2}P(T)P(H);P_n=\frac{P_{n-1}}{2}+\frac{P_{n-2}}{4};P_1=1,P_2=\frac{3}{4};P_3=\frac{3}{8}+\frac{1}{4}$
$P_3=\frac{5}{8};P_4=\frac{P_3}{2}+\frac{P_2}{4};=\frac{8}{16}$