At a conference, there are a total of 275 attendees. Each attendee is assigned to either group A, group B,...

1. TRANSLATE the problem information

• Given information:
  • Total attendees: 275 people
  • Probability of selecting Group A member: 0.44
  • Probability of selecting Group B member: 0.24
  • Each person is assigned to exactly one group (A, B, or C)

• What we need to find: Number of people in Group C

2. INFER the key relationship

• Since every attendee is assigned to exactly one group, the probabilities of selecting from each group must add up to 1
• This means: \(\mathrm{P(Group\,A) + P(Group\,B) + P(Group\,C) = 1}\)
• We can use this to find \(\mathrm{P(Group\,C)}\)

3. SIMPLIFY to find P(Group C)

• \(\mathrm{P(Group\,C) = 1 - P(Group\,A) - P(Group\,B)}\)
• \(\mathrm{P(Group\,C) = 1 - 0.44 - 0.24 = 0.32}\)

4. TRANSLATE probability to actual count

• If \(\mathrm{P(Group\,C) = 0.32}\), then 32% of all attendees are in Group C
• Number in Group C = \(\mathrm{0.32 \times 275 = 88}\) people

Answer: 88

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER reasoning: Students don't recognize that the probabilities must sum to 1. Instead, they might try to work with the given probabilities directly without finding \(\mathrm{P(Group\,C)}\) first.

For example, they might incorrectly calculate: \(\mathrm{275 \times 0.44 = 121}\), then \(\mathrm{275 \times 0.24 = 66}\), then think the answer is \(\mathrm{121 + 66 = 187}\). This shows they're confusing what the question is asking and not using the complement relationship.

This leads to confusion and incorrect calculations.

Second Most Common Error:

Incomplete SIMPLIFY execution: Students correctly identify that \(\mathrm{P(Group\,C) = 1 - 0.44 - 0.24}\), but make arithmetic errors in the subtraction or the final multiplication.

Common calculation mistakes include:

• Getting \(\mathrm{P(Group\,C) = 0.22}\) instead of 0.32
• Multiplying incorrectly: \(\mathrm{0.32 \times 275 \neq 88}\)

This may lead them to select an incorrect numerical answer.

The Bottom Line:

This problem tests whether students understand that probabilities in a complete system must sum to 1. The key insight is recognizing the complement relationship rather than trying to work with incomplete information.