A playlist contains 15 distinct songs. The playlist is shuffled uniformly at random, and the first song is played. What...

1. TRANSLATE the problem information

• Given information:
  • 15 distinct songs in playlist
  • Playlist is shuffled uniformly at random
  • We want probability that Song T is played first

• What this tells us: We need to find \(\mathrm{P(Song\;T\;is\;in\;first\;position)}\)

2. INFER the probability structure

• Key insight: 'Uniformly at random' means every possible arrangement of the 15 songs is equally likely
• Since all arrangements are equally likely, each individual song has the same chance of ending up in the first position
• This becomes a simple equal-outcomes probability problem

3. INFER the calculation approach

• Total possible songs that could be first: 15 (any of the 15 songs)
• Favorable outcomes: 1 (only Song T satisfies our condition)
• Since each song is equally likely to be first: \(\mathrm{P(Song\;T\;first)} = \frac{1}{15}\)

Answer: A. \(\frac{1}{15}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students misinterpret what happens after the first song is selected, thinking there are only 14 songs left to consider for the probability calculation.

They incorrectly reason: 'After Song T is selected, there are 14 remaining songs, so the probability should be 1/14.' This confuses the probability of Song T being first with some other conditional probability scenario.

This may lead them to select Choice B (\(\frac{1}{14}\)).

The Bottom Line:

This problem tests whether students can recognize a basic equal-probability situation. The key insight is that when distinct objects are arranged randomly, each object has the same probability of occupying any specific position - in this case, 1 out of 15 for the first position.