A certain coin lands head with probability p. Let Q denote the probability that when the coin is tossed four times the number of heads obtained is even. Then

Question

A certain coin lands head with probability p. Let Q denote the probability that when the coin is tossed four times the number of heads obtained is even. Then (A) there is no value of p, if Q =(1/4) (B) there is exactly one value of P if Q =(3/4) (C) there are exactly two values of p if Q =(3/5) (D) there are exactly four values of p if Q =(4/5)

Tardigrade Admin · Accepted Answer

P (0 H or 2 H or 4 H ) Q=p4+6 p2(1-p)2+(1-p)4=8 p4-16 p3+12 p2-4 p+1=((2 p-1)4+1/2). Now interpret

Q. A certain coin lands head with probability $p$ . Let $Q$ denote the probability that when the coin is tossed four times the number of heads obtained is even. Then

there is no value of $p$ , if $Q = \frac{1}{4}$

there is exactly one value of $P$ if $Q = \frac{3}{4}$

there are exactly two values of $p$ if $Q = \frac{3}{5}$

there are exactly four values of $p$ if $Q = \frac{4}{5}$

$P (0 H$ or $2 H$ or $4 H)$
$Q = p^{4} + 6 p^{2} (1 - p)^{2} + (1 - p)^{4} = 8 p^{4} - 16 p^{3} + 12 p^{2} - 4 p + 1 = \frac{( 2 p - 1 ) ^{4} + 1}{2}$ . Now interpret

Q. A certain coin lands head with probability p. Let Q denote the probability that when the coin is tossed four times the number of heads obtained is even. Then

there is no value of p, if Q=41​

there is exactly one value of P if Q=43​

there are exactly two values of p if Q=53​

there are exactly four values of p if Q=54​