If I have 99 songs in a music playlist, what are the chances of pressing shuffle and seeing the same song first, 4 times consecutively?

You didn't need to say that it had to be a specific song. You just want the probability that whatever plays the first time, it is repeated 3 more times.

- The first time you press shuffle, you get a song. That'll be the one you need to match.

- The second time, the probability of getting that same song is 1/99

- The third time, the probability of getting that same song is 1/99

- The fourth time, the probability of getting that same song is 1/99

So the overall probability of hearing the same song 4 times is:

1/99 * 1/99 * 1/99

≈ 1/970299

If we were to change it to 100 songs, it would be:

1/100 * 1/100 * 1/100

= 1/1000000

In other words, it's close to 1 in a million.

I would question whether the shuffle play is truly random. I'd also wonder if there are really that many songs in the playlist or perhaps that the algorithm isn't considering all the songs when it starts.

Let the first song be what ever it is. 

There is 1/99 chance that it occurs first again.

Then 1/99 again.

Then 1/99 again.

Making a total of (1/99)^3.

Now a slightly different question is, what if you want a particular song chosen first.

There is a 1/99 chance that it is the first one. Thus a total chance of (1/99)^4. 

1/99 mutiplied by itself 4 times.