# Probability of a desired song being played at max the 4th song.

I have an MP3 player and I loaded my Mp3 player with 3 songs from Michael jackson, 5 songs from Cardi B, 5 songs from Snoop Dog and 7 songs from Korn and I play them in shuffe mode (every song is played once in random order). What is the probability that the first song by Michael Jackson is played at the lastest as the fourth song?

So I think I should use this formula $$P[A]= \sum_{i=4}^n P[A|B_i]P[B_i] = (3/20)+(3/19)*(17/18)+(3/18)(17/18)(16/17)....$$ And so on. Is this the right track? Would love it if someone could provide some hints.

• A given song has an equal chance of being in any slot. Thus your chosen song will be amongst the first four with probability $\frac 4{20}$. – lulu Mar 27 '18 at 12:38
• @lulu: There is no chosen song, though. The problem asks for the probability that there is at least one MJ song (any of the three) in the first four shuffle selections. – Brian Tung Mar 27 '18 at 21:04
• I had to laugh a little because Korn is nothing like the other three artists, but I'm glad they are being represented!!! – WaveX Mar 27 '18 at 21:14
• @BrianTung Ah, I read the question as suggesting that the MJ songs are numbered. I agree, your reading is more probable. – lulu Mar 27 '18 at 21:23
• @WaveX: Neither is MJ, for that matter. I know very little about Cardi B, so can't say how similar she is to Snoop. – Brian Tung Mar 27 '18 at 21:28

## 1 Answer

The probability that the first song by Michael Jackson (in the shuffle ordering) is one of the first four selections is $1$ minus the probability that there aren't any of them in the first four selections; that is,

\begin{align} P(\text{the first} & \text{ four selections include an MJ song}) \\ & = 1-P(\text{the first four selections include no MJ songs}) \\ & = 1-P(\text{the three MJ songs are all in the last $16$ selections}) \\ & = 1-\frac{\text{the number of ways to choose three of the last $16$ selections}}{\text{the number of ways to choose three of all $20$ selections}} \end{align}

Does that help?

Your approach can also work, but I don't understand where you came up with the numbers you use. It seems to me that it should be

\begin{align} P(\text{the first} & \text{ four selections include an MJ song}) \\ & = P(\text{the first selection is an MJ song}) \\ & + P(\text{the first selection isn't an MJ song, but the second is}) \\ & + P(\text{the first two selections aren't MJ songs, but the third is}) \\ & + P(\text{the first three selections aren't MJ songs, but the fourth is}) \\ & = \frac{3}{20} + \frac{17}{20} \times \frac{3}{19} + \frac{17}{20} \times \frac{16}{19} \times \frac{3}{18} + \frac{17}{20} \times \frac{16}{19} \times \frac{15}{18} \times \frac{3}{17} \end{align}

in which case you should end up with the same answer yielded by the first analysis.

Unless I completely misunderstand the question.