A park has 7 beds of roses and 6 beds of tulips. Each rose bed has 10 red flowers and...

1. TRANSLATE the problem information

• Given information:
  • 7 rose beds with 10 red + 8 white flowers each
  • 6 tulip beds with 11 red + 9 white flowers each
  • Need: \(\mathrm{P(Tulip\ flower | Red\ flower)}\)

• What this tells us: We need conditional probability - given the flower is red, what's the chance it's a tulip?

2. INFER the approach

• This is conditional probability: \(\mathrm{P(Tulip | Red) = \frac{Red\ tulips}{Total\ red\ flowers}}\)
• Since we're told the flower IS red, we only consider red flowers in our calculation
• We need to count red tulips and total red flowers

3. Calculate the counts

• Red tulips: \(\mathrm{6\ beds × 11\ red\ per\ bed = 66\ red\ tulips}\)
• Red roses: \(\mathrm{7\ beds × 10\ red\ per\ bed = 70\ red\ roses}\)
• Total red flowers: \(\mathrm{66 + 70 = 136\ red\ flowers}\)

4. SIMPLIFY to find the probability

• \(\mathrm{P(Tulip | Red) = \frac{66}{136}}\)
• Find GCD: \(\mathrm{66 = 2 × 33, 136 = 2 × 68, GCD = 2}\)
• Simplified: \(\mathrm{\frac{66}{136} = \frac{33}{68}}\)

Answer: C (33/68)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret "probability of tulip given red" and instead calculate the overall probability of selecting a tulip flower from all flowers.

They calculate: \(\mathrm{\frac{Total\ tulips}{Total\ flowers} = \frac{120}{246}}\), which doesn't match any answer choice. This leads to confusion and guessing.

Second Most Common Error:

Poor INFER reasoning: Students don't recognize they need to focus on beds as units and only count 1 bed of each type.

They calculate: \(\mathrm{P(Tulip | Red) = \frac{11}{10+11} = \frac{11}{21}}\), leading them to select Choice D (11/21).

The Bottom Line:

Conditional probability problems require careful attention to what information restricts your sample space - here, knowing the flower is red means you only consider red flowers, not all flowers.