A magician has $5$ coins. He initially places $3$ of the coins with heads-up and the rest with tails-up. Then he performs a process in which he flips one coin every second. The process stops when all the coins are tails-up. What is the probability that the process ends in $3$ seconds.
I found two methods two solve this but both are giving different answers.
Method 1: The process will end in exactly $3$ seconds when at each step the the coin with heads-up is flipped.
The probability of choosing one heads-up coin in first step is $\frac{3}{5}$.
Now we have flipped one heads-up coin. So, probability of choosing one heads-up coin in second step is $\frac {2}{5}$.
Similarly in third step it is $\frac{1}{5}$.
So, the probability that the process ends in three seconds is $(\frac{1}{5} \cdot \frac{2}{5} \cdot \frac{3}{5})=\frac{6}{125}$
Method 2: The process will end in $3$ seconds if we flip heads-up coin in all the steps.
We wi ll find all the possible sequences of steps:
$HHH$
$HHT$
$HTH$
$THH$
$HTT$
$TTH$
$THT$
Where $H$ or $T$ at the $i_{th}$ position represents the heads-up or tails-up coin respectively flipped at $i_{th}$ second.
So, the required probability is $\frac{1}{7}$
Why am I getting different answers? Which one is wrong?
Method $1$ is correct. Method $2$ is completely wrong: the probability of picking a heads-up coin to flip changes with every step.