At an end-of-year party for the employees of a certain company, Martha has been given an opportunity to win as...
GMAT Two Part Analysis : (TPA) Questions
At an end-of-year party for the employees of a certain company, Martha has been given an opportunity to win as many as two prizes. She must draw, at random, exactly one numbered card from each of two boxes—each of which, in turn, contains exactly six cards, numbered from \(1\) to \(6\). If the sum of the two numbers is equal to \(8\), then Martha will win a tablet computer. And, if at least one of the two numbers is a \(2\), then Martha will win a smartphone.
Select the probability that Martha will win a Tablet computer, and select the probability that Martha will win Both prizes. Make only two selections, one in each column.
Phase 1: Owning the Dataset
Visualization
Let's create a 6×6 grid showing all possible outcomes:
| First Box ↓ | Second Box → 1 | 2 | 3 | 4 | 5 | 6 |
| 1 | (1,1) | (1,2) | (1,3) | (1,4) | (1,5) | (1,6) |
| 2 | (2,1) | (2,2) | (2,3) | (2,4) | (2,5) | (2,6) |
| 3 | (3,1) | (3,2) | (3,3) | (3,4) | (3,5) | (3,6) |
| 4 | (4,1) | (4,2) | (4,3) | (4,4) | (4,5) | (4,6) |
| 5 | (5,1) | (5,2) | (5,3) | (5,4) | (5,5) | (5,6) |
| 6 | (6,1) | (6,2) | (6,3) | (6,4) | (6,5) | (6,6) |
Total possible outcomes: \(6 \times 6 = 36\)
Key Conditions
- Tablet Computer: Sum of two cards = 8
- Smartphone: At least one card shows 2
Phase 2: Understanding the Question
We need to find:
- Probability of winning Tablet Computer
- Probability of winning Both prizes
Phase 3: Finding the Answer
Part 1: Tablet Computer (Sum = 8)
Let's identify all pairs that sum to 8:
- (2,6): \(2 + 6 = 8\) ✓
- (3,5): \(3 + 5 = 8\) ✓
- (4,4): \(4 + 4 = 8\) ✓
- (5,3): \(5 + 3 = 8\) ✓
- (6,2): \(6 + 2 = 8\) ✓
That's 5 favorable outcomes out of 36 total.
Probability of Tablet = \(\frac{5}{36}\)
Part 2: Both Prizes
For both prizes, we need:
- Sum = 8 (for tablet) AND
- At least one 2 (for smartphone)
From our five pairs that sum to 8, which ones contain at least one 2?
- (2,6): Contains a 2 ✓
- (3,5): No 2
- (4,4): No 2
- (5,3): No 2
- (6,2): Contains a 2 ✓
Only 2 outcomes satisfy both conditions.
Probability of Both = \(\frac{2}{36} = \frac{1}{18}\)
Phase 4: Solution
Final Answer
- Tablet Computer: \(\frac{5}{36}\)
- Both prizes: \(\frac{1}{18}\)