Shuffling Songs probability question

[Kyuutoryuu]

Hello. I have the following problem:

Let's say you have 354 songs in your entire music library. And let's say you started playing those songs, in shuffle mode, starting from the very first (random) song. Assuming perfect randomization, let's say you are now on the 34th random song in your shuffle. If you have one specific favorite song in your entire library, what is the probability that your specific favorite song is selected either as the 34th song or earlier?

I calculated the probability as follows:

(1/354) + (353/354)(1/353) + (353/354)(352/353)(1/352) + ... + (353/354)(352/353)(...)(325/326)(1/325)

Simplification of terms leads to (1/354) + (1/354) + (1/354) + ... + (1/354)

..which in turn leads to 34(1/354) = (34/354).Was this probability value calculated correctly?

(In case you're wondering, the method I used was to add all of the probabilities of selecting your favorite song at each specific shuffle count. Each grouping of parentheses represent each individual probability, and the third grouping, for example, was the probability that the favorite song did not come up on either of the first two songs but did come up on the third song).

 

[mathman]

The probability that it has not been selected is (353/354)(352/353)...(320/321)=(320/354) which gives the same result you got.

I believe it is clearer this way.

 

[paisiello2]

Usually these kind of problems are solved by calculating the probability of the event not happening and then subtracting this number from 100%.

 

[mfb]

The calculation is correct, but as you can guess from the result, it is way too complicated.

The probability that your song is in one of the first 34 places of 354, for a perfectly random distribution, is 34/354.

 

[Martin Rattigan]

This assumes that a shuffle is choosing one of the 354! orderings at random (analogous to shuffling a pack of cards) when mfb's comments are obviously the way to do it. The remarks about calculating the probability of not happening apply when the songs are chosen at random from the whole set for each play when the probability is [itex]1-(1-\frac{1}{354})^{34}[/itex] which is only approximately [itex]\frac{34}{354}[/itex].