Undestanding Memorylessness and Geometirc Random Variables

Question

I know there is a lot on internet about it but I'm still not clear about the concept.Suppose a person starts tossing coins right now.After 4 tosses(all tail) another person enters the room.By independence,the person who enters know given that first 4 tosses were tail see the experiment no differently from the one who was tossing before.Now say the next toss is a head.According to first gut Pr(success on fifth trial) = (1/2)**5 whereas according to second guy = 1/2.Who is corect?

Answer 1

The idea of memoryless is that if $X$ is geometrically distributed, then $$ P(X = n | X \ge m) = P(X = n - m), $$ where $n \ge m$. As you state, the chance of flipping $t,t,t,t,h$ equals $(\frac{1}{2})^5$, so the first guy is right. However, the chance of throwing heads at the fifth flip given that you have thrown four tails equals $\frac{1}{2}$. Note that those are different events.

Answer 2

Both, only they are not computing the same thing: first guy is considering Pr(first success on fifth trial)=(1/2)**5 while second guy is considering Pr(first sucess on fifth trial|no success in first four trials)=Pr(first success on first trial)=(1/2).