A fair coin is flipped six times. What is the probability that heads occurs exactly 6 times if it is known that heads occurs at least four times?

Question

Just as the title says.

 

A fair coin is flipped six times. What is the probability that heads occurs exactly 6 times if it is known that heads occurs at least four times?

Answer 1

The probability that heads occurs exactly 6 times if it is known that heads occurs at least four times is

P{H=6|H≥4}

= P{(H=6)∩(H≥4)}/P{H≥4}

= P{H=6}/P{H≥4}               [since (H=6) ⊂ (H≥4)]

 

where

P{H=6} = C(6,6)(½)⁶

            = (½)⁶

 

P{H≥4} = P{H=4} + P{H=5} + P{H=6}

            = C(6,4)(½)⁴(½)² + C(6,5)(½)⁵(½) + C(6,6)(½)⁶

            = (½)⁶[C(6,4) + C(6,5) + C(6,6)]

            = (½)⁶[15 + 6 + 1]

            = (½)⁶(22)

 

Therefore,

P{H=6|H≥4} = 1/22 = 0.04545