A biased coin with probability p, `0ltplt1` of heads is tossed until a head appears for the first time. If the probability that the number of tosses r

Question

A biased coin with probability p, 0<p<1 of heads is tossed until a head appears for the first time. If the probability that the number of tosses required is even is 2/5, then p equals A. 2/3 B. 1/2 C. 1/3 D. 1/4 Select the correct answer from above options

Answer 1

Correct Answer - C The number of tosses required to get head is even means that the head is obtain in either 2 n d 2 toss or in 4th toss or in 6th toss etc. ∴ ( 1 − p ) p + ( 1 − p ) 3 p + ( 1 − p ) 5 p + . . = 2 5 ∴ ⇒ ( 1 − p ) p 1 − ( 1 − p ) 2 = 2 5 ⇒ 5 p ( 1 − p ) = 2 ( 2 p − p 2 ) ⇒ ⇒ 3 p 2 − p = 0 ⇒ p = 1 3 ⇒