A box contains 13 red pens and 37 blue pens. If one of these pens is selected at random, what...

1. TRANSLATE the problem information

• Given information:
  • 13 red pens in the box
  • 37 blue pens in the box
  • One pen selected at random
• What we need: Probability of selecting a red pen

2. INFER what probability means in this context

• Probability = Number of favorable outcomes / Total number of possible outcomes
• Favorable outcomes = red pens = 13
• Total outcomes = all pens in the box = red pens + blue pens

3. Calculate the total number of pens

• Total pens = \(13 + 37 = 50\) pens

4. SIMPLIFY to find the probability

• \(\mathrm{P(red\:pen)} = \frac{13}{50}\)
• Converting to decimal: \(13 \div 50 = 0.26\)

Answer: \(\frac{13}{50}\) or \(0.26\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students misunderstand what goes in the denominator of a probability fraction.

Instead of using total outcomes (all pens), they might use just the red pens in both numerator and denominator, getting \(\frac{13}{13} = 1\). Or they might think they need to compare red pens to blue pens directly, getting \(\frac{13}{37}\). This leads to confusion and incorrect calculations that don't represent probability at all.

Second Most Common Error:

Poor attention to output format: Students correctly calculate \(\frac{13}{50} = 0.26\) but convert to percentage form anyway.

They calculate \(0.26 \times 100 = 26\%\) and submit "26%" despite the problem explicitly stating "not as a percent." This leads them to give an answer in the wrong format even though their mathematical reasoning was correct.

The Bottom Line:

This problem tests whether students truly understand the basic probability model. The calculation itself is straightforward, but students must correctly identify what constitutes the "universe" of all possible outcomes (all 50 pens) versus just the favorable outcomes (13 red pens).