A fair coin is flipped six times. What is the probability that heads occurs exactly 3 times if it is known that heads occurs at least three times?

Answer 1

6C3(1/2)^6

= 6!/3!3!(1/2^6)

=20/64= 0.3125 = unconditional probability of exactly 3 heads,

6C4(1/2)^6

= 6!/4!2!(1/64)

= 15/64 = 0.234375= unconditional probability of exactly 4 heads

6C5(1/2)^6

= 6/64 = 3/32 = 0.09375 = unconditional probability of exactly 5 heads

6C6(1/2)^6

= 1/64 = 0.015625 = unconditional probability of exactly 6 heads

sum them = (20+15+6+1)/64 = 42/64 =0.65625

then find the conditional probability of 3 heads given at least 3 heads

=0.3125/0.65625 = (20/64)/(42/64) = 20/42 = 10/21 =

= 0.476190476

= about 47.62% chance of 3 heads conditioned on at least 3 heads

an alternative method is

calculate 1- P(0)-P(1)-P(2)= P(<3 heads)

= 1 - 1/64 - 6/64 -15/64

= 1 - 21/64 = (64-22)/64

= 42/64

divide P(3) by 42/64

= 20/64 divided by 42/64

= about 47.62%