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 shuffled uniformly at random
  • Want probability that Song T is played first
• What "uniformly at random" means: All possible arrangements of the 15 songs are equally likely

2. INFER the key insight

• Since the shuffle is uniform and random, each individual song has the same chance of ending up in the first position
• This is a symmetry situation - no song is "favored" over any other
• We don't need to count complex arrangements; we just need to think about Song T's equal share

3. Apply the basic probability approach

• Favorable outcomes: 1 (Song T is first)
• Total possible outcomes for first position: 15 (any of the 15 songs could be first)
• Probability = \(\frac{1}{15}\)

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

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students overthink the problem and attempt to use permutation formulas unnecessarily

Instead of recognizing the simple symmetry, they might think: "I need to count all possible arrangements where Song T is first (\(14!\)) and divide by total arrangements (\(15!\))." While this approach actually works and gives the correct answer, it's unnecessarily complex and prone to calculation errors.

Some students get confused in the permutation setup and make errors like using \(\frac{15!}{14!}\) instead of \(\frac{14!}{15!}\), which could lead them to select Choice (B) \(\frac{1}{14}\).

The Bottom Line:

This problem tests whether students can recognize when a complex-looking probability situation actually has a simple, intuitive solution based on symmetry. The key insight is that uniform random shuffling treats all songs equally.