Human blood can be classified into four common blood types—A, B, AB, and O. It is also characterized by the...
GMAT Problem-Solving and Data Analysis : (PS_DA) Questions
Human blood can be classified into four common blood types—A, B, AB, and O. It is also characterized by the presence or absence of the rhesus factor.
The table above shows the distribution of blood type and rhesus factor for a group of people. If one of these people who is rhesus negative is chosen at random, the probability that the person has blood type B is \(\frac{1}{9}\). What is the value of x?
| Rhesus factor | Blood type | |||
|---|---|---|---|---|
| A | B | AB | O | |
| + | 33 | 9 | 3 | 37 |
| - | 7 | 2 | 1 | x |
1. TRANSLATE the problem information
- Given information:
- Table shows blood type distribution by rhesus factor
- Rhesus negative people: \(\mathrm{A=7, B=2, AB=1, O=x}\) (unknown)
- Probability that a randomly chosen rhesus negative person has blood type B is \(\frac{1}{9}\)
- What this tells us: We need to set up a probability equation using the rhesus negative subset only
2. INFER the approach
- Since we want the probability within the rhesus negative group, we need:
- Total rhesus negative people as our denominator
- Number of rhesus negative people with blood type B as our numerator
- This creates a probability equation we can solve for x
3. SIMPLIFY by finding the total rhesus negative population
- Total rhesus negative people = \(\mathrm{7 + 2 + 1 + x = 10 + x}\)
- Rhesus negative people with blood type B = 2
4. TRANSLATE the probability condition into an equation
- Probability = favorable outcomes/total outcomes
- \(\frac{2}{10 + \mathrm{x}} = \frac{1}{9}\)
5. SIMPLIFY to solve for x
- Cross multiply: \(\mathrm{2 × 9 = 1 × (10 + x)}\)
- \(\mathrm{18 = 10 + x}\)
- \(\mathrm{x = 8}\)
Answer: 8
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students may misinterpret "rhesus negative" and include all people in their probability calculation, not just the rhesus negative subset.
They might calculate probability as \(\frac{2}{\mathrm{total\ of\ all\ people}} = \frac{2}{33+9+3+37+7+2+1+\mathrm{x}}\) instead of \(\frac{2}{10+\mathrm{x}}\). This leads to a much more complex equation that doesn't yield a clean integer answer, causing confusion and potentially leading to guessing.
Second Most Common Error:
Poor SIMPLIFY execution: Students correctly set up \(\frac{2}{10 + \mathrm{x}} = \frac{1}{9}\) but make algebraic errors in solving.
Common mistake: Instead of cross multiplying to get \(\mathrm{18 = 10 + x}\), they might multiply both sides by \(\mathrm{(10 + x)}\) incorrectly or forget to distribute properly. This leads to wrong values of x that don't match the given answer.
The Bottom Line:
This problem tests your ability to work with conditional probability - understanding that "chosen from those who are rhesus negative" means your sample space is only the rhesus negative people, not everyone in the table. The key insight is recognizing this constraint before setting up your probability equation.