Question:The 11 letters of the word PROBABILITY are each printed on a separate tile, with each tile containing exactly one...

1. TRANSLATE the problem information

• Given information:
  • The word PROBABILITY has 11 letters printed on separate tiles
  • We're drawing one tile at random
  • We want the probability that the drawn tile has a letter appearing more than once in the word
• What this tells us: We need to find letters that appear multiple times, then count how many tiles contain those letters.

2. INFER the approach

• We need to count the frequency of each letter in PROBABILITY
• Identify which letters appear more than once (frequency \(\geq 2\))
• Count the total number of tiles containing those letters
• Apply the probability formula

3. Count letter frequencies systematically

Let's go through PROBABILITY letter by letter:
P-R-O-B-A-B-I-L-I-T-Y

Counting each letter:

• P: appears 1 time
• R: appears 1 time
• O: appears 1 time
• B: appears 2 times
• A: appears 1 time
• I: appears 2 times
• L: appears 1 time
• T: appears 1 time
• Y: appears 1 time

4. TRANSLATE "more than once" requirement

Letters that appear more than once (frequency \(\gt 1\)):

• B appears 2 times
• I appears 2 times

5. Count favorable outcomes

• Total number of tiles containing letters that appear more than once:
  • 2 tiles with letter B
  • 2 tiles with letter I
  • Total favorable outcomes \(= 2 + 2 = 4\)

6. Apply probability formula

• Total possible outcomes = 11 tiles
• Favorable outcomes = 4 tiles
• Probability \(= \frac{4}{11}\)

Answer: B) \(\frac{4}{11}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret what constitutes a "favorable outcome"

They count the number of different letters that appear more than once (B and I = 2 letters) instead of counting the actual number of tiles containing those letters (4 tiles total). This leads to calculating \(\frac{2}{11}\) as the probability.

This may lead them to select Choice A (\(\frac{2}{11}\))

Second Most Common Error:

Poor frequency counting: Students miscount how many times certain letters appear in PROBABILITY

Common mistakes include missing one of the B's or I's, or incorrectly thinking other letters appear multiple times. This leads to identifying the wrong letters as appearing "more than once" or counting the wrong number of favorable tiles.

This causes them to get stuck and guess among the remaining answer choices.

The Bottom Line:

This problem requires careful attention to the distinction between counting unique letters versus counting individual tiles. The key insight is that when B appears twice, both B tiles are favorable outcomes, not just counting B as "one letter that appears multiple times."