there is a long and short way to answer this.
The most laborious way is as follows...
If one of the first 4 shirts is a medium, then....
A. the first one is a medium and the other 3 are not
the probability of this is
B. the 2nd one is medium and the other 3 are not
the probability is
which is in fact the same as A.
then there is the probability that the 3rd one is medium and the other 3 are not....
and there is the probability that the first 3 are not medium and the 4th one is.
Having done that, you now need to work out the probabilty of 2 of the 4 being medium,
the probability of 3 of the 4 being medium,
the probability of all four being medium.
There is a shortcut... thankfully.
Since all probabilities sum to one,
it's only necessary to calculate the probability of all 4 shirts not being medium
and subtract the answer from 1.
Therefore the probability that at least one shirt is a medium is
Your friend calculated the following...
If at least 1 medium shirt is picked with 4 attempts,
then there is a probability of picking it at the first try.
Should this happen, we don't need to care what happens next.
There is a probability
that the first medium is chosen at the 2nd attempt, we don't need to care what happens next.
There is a
probability of picking the 1st medium shirt on the 3rd go.
There is a
probability the 1st medium shirt is chosen on the 4th try.
Sum those probabilities....
That is also correct!
The answer is 62.435500516%