Question

A fair coin is tossed 10 times. Let p be the probability of getting a head in the hird trial (but not before) and let q be the probability of getting the head after the second trial (but not in the first two trials) then p q is very close to a) 1 4 b) 1 2 c) 2 d) 4 Select the correct answer from above options

Answer 1

Here. p is the probability of getting a head in third trial. It means, first two trials will be tails and third trial will be head. ∴p=P(T)⋅P(T)⋅P(H)= 1 2 ⋅ 1 2 ⋅ 1 2 = 1 8 Now, q is the probability of getting heads after two trials. It can be heads after two trials, third trials and so on. ∴q=( 1 2 ⋅ 1 2 ⋅ 1 2 )+( 1 2 ⋅ 1 2 ⋅ 1 2 ⋅ 1 2 )+( 1 2 ⋅ 1 2 ⋅ 1 2 ⋅ 1 2 ⋅ 1 2 )+...( 1 2 ⋅ 1 2 ⋅ 1 2 ⋅ 1 2 ⋅ 1 2 ⋅ 1 2 ⋅ 1 2 ⋅ 1 2 ⋅ 1 2 ⋅ 1 2 ⋅) ⇒q= 1 8 + 1 16 +... 1 1024 It is a G.P. with first term, a= 1 8 and common ratio, r= 1 2 with total terms 8. ∴q= 1 8 ( 1-( 1 2 )8 1- 1 2 ) ∴ p q = 1 8 1 8 ( 1-( 1 2 )8 1- 1 2 ) ⇒ p q = 1- 1 2 1- 1 256 = 1 2 255 256 = 128 255 So, p q is close to 1/2.