According to a prominent investment adviser, Company X has a 50% chance of posting a profit in the coming year,...

Phase 1: Owning the Dataset

Let's visualize this problem to make it crystal clear. We have two companies with different profit probabilities, and we need to find the range of possible joint probabilities.

Concrete Numbers

Let's imagine 100 possible future scenarios:

• Company X profits in 50 scenarios (50%)
• Company Y profits in 60 scenarios (60%)

Visual Representation

We'll use overlapping circles (Venn diagram concept) to show the probability space:

Total probability space = 100%
┌─────────────────────────────────┐
│                                 │
│    ┌─────────┐                  │
│    │    X    │ ┌─────────┐      │
│    │  (50%)  │ │    Y    │      │
│    │ ┌───────┼─┤  (60%)  │      │
│    └─┤Overlap│ └─────────┘      │
│      └───────┘                  │
└─────────────────────────────────┘

The overlap represents scenarios where BOTH companies profit.

Phase 2: Understanding the Question

Breaking Down the Question

We need to find:

1. Least probability: The minimum possible overlap between X's profitable scenarios and Y's profitable scenarios
2. Greatest probability: The maximum possible overlap between X's profitable scenarios and Y's profitable scenarios

Key Mathematical Principle

The joint probability \(\mathrm{P(X \text{ and } Y)}\) is constrained by:

• Lower bound: \(\mathrm{max}(0, \mathrm{P(X)} + \mathrm{P(Y)} - 1)\)
• Upper bound: \(\mathrm{min}(\mathrm{P(X)}, \mathrm{P(Y)})\)

This makes intuitive sense:

• We can't have negative overlap (hence max with 0)
• The overlap can't exceed either individual probability

Phase 3: Finding the Answer

Calculating the Least Probability

When do X and Y have minimal overlap? When they're as separate as possible.

Calculation:

• \(\mathrm{P(X)} + \mathrm{P(Y)} = 50\% + 60\% = 110\%\)
• Since this exceeds 100%, they MUST overlap by at least: \(110\% - 100\% = 10\%\)

Visual for minimum overlap:

100% total space
├─── X only (40%) ───┼─ Both (10%) ─┼─── Y only (50%) ───┤
└────── X (50%) ─────┴────────────── Y (60%) ───────────┘

Least probability = 10% ✓

Calculating the Greatest Probability

When do X and Y have maximal overlap? When one is completely contained within the other.

Since \(\mathrm{X\ (50\%)} < \mathrm{Y\ (60\%)}\), all of X's profitable scenarios could be contained within Y's profitable scenarios.

Visual for maximum overlap:

100% total space
├─────────── Y (60%) ────────────┼─── Neither (40%) ───┤
└─── X=Both (50%) ───┼─Y only(10%)┤

Greatest probability = 50% ✓

Phase 4: Solution

Final Answer:

• Least Probability for Both: 10%
• Greatest Probability for Both: 50%

These answers emerge directly from our probability constraints:

• The 10% minimum occurs when the companies' profits overlap as little as mathematically possible
• The 50% maximum occurs when Company X's profits are entirely contained within Company Y's profits