For a certain set of carnival games (Games 1-6), the probability of a game being won is perfectly negatively correlated...
GMAT Two Part Analysis : (TPA) Questions
For a certain set of carnival games (Games 1-6), the probability of a game being won is perfectly negatively correlated to the value of the reward for winning. Game 1 has the second-highest probability of being won. Game 2 has the second-lowest probability of being won. The value of the rewards for Games 4 and 6 are between the values for Games 1 and 2. Game 5 has a lower-value reward than does Game 2. No two games have the same probability of being won.
Based on the information provided and from among the six carnival games, select for Highest the game with the highest probability of being won, and select for Lowest the game with the lowest probability of being won. Make only two selections, one in each column.
Phase 1: Owning the Dataset
Since we have a ranking problem with perfect negative correlation, let's use a comparison table to track probabilities and rewards.
| Game | Probability Rank | Reward Rank |
| Game 1 | 2nd highest | ? |
| Game 2 | 2nd lowest (5th) | ? |
| Game 3 | ? | ? |
| Game 4 | ? | Between Games 1 & 2 |
| Game 5 | ? | Lower than Game 2 |
| Game 6 | ? | Between Games 1 & 2 |
Key insight: Perfect negative correlation means if a game has the nth highest probability, it has the nth lowest reward (or \(7-\mathrm{n}\) highest reward).
Phase 2: Understanding the Question
We need to find:
- Highest: The game with the highest probability of being won
- Lowest: The game with the lowest probability of being won
Since probability and reward are perfectly negatively correlated:
- Game 1 (2nd highest probability) → 2nd lowest reward (5th in reward ranking)
- Game 2 (2nd lowest probability = 5th in probability ranking) → 2nd highest reward
Let's update our table:
| Game | Probability Rank | Reward Rank |
| Game 1 | 2nd | 5th |
| Game 2 | 5th | 2nd |
| Game 3 | ? | ? |
| Game 4 | ? | 3rd or 4th |
| Game 5 | ? | ? |
| Game 6 | ? | 3rd or 4th |
Phase 3: Finding the Answer
Games 4 and 6 have rewards between Game 1 (5th) and Game 2 (2nd), so they must occupy the 3rd and 4th reward positions.
Game 5 has a lower reward than Game 2 (2nd highest). The available positions are:
- 1st (highest) - Not taken yet
- 3rd or 4th - Taken by Games 4 and 6
- 5th - Taken by Game 1
- 6th (lowest) - Not taken yet
Since Game 5 has lower reward than Game 2, it can't be 1st. Since 3rd, 4th, and 5th are taken, Game 5 must have the 6th (lowest) reward.
With perfect negative correlation:
- Game 5 (6th/lowest reward) → 1st/highest probability
- By elimination, Game 3 must have 1st/highest reward → 6th/lowest probability
Final ranking:
| Game | Probability Rank | Reward Rank |
| Game 1 | 2nd | 5th |
| Game 2 | 5th | 2nd |
| Game 3 | 6th (lowest) | 1st (highest) |
| Game 4 | 3rd or 4th | 4th or 3rd |
| Game 5 | 1st (highest) | 6th (lowest) |
| Game 6 | 4th or 3rd | 3rd or 4th |
Phase 4: Solution
Highest probability of being won: Game 5
Lowest probability of being won: Game 3