Events A, B, and C have the following probabilities of future occurrence: 20% 50% 80%...

Phase 1: Owning the Dataset

Visualization

Let's create a probability scale to understand the given information:

0% -------- 20% -------- 50% -------- 80% -------- 100%
             A            B             C

Given probabilities:

• Event A: 20%
• Event B: 50%
• Event C: 80%

Phase 2: Understanding the Question

We need to find:

1. Both A and B: A description that MUST be true of \(\mathrm{P(A \text{ and } B)}\)
2. B or C or Both: A description that MUST be true of \(\mathrm{P(B \text{ or } C)}\)

Key Insight for "Both A and B"

The probability of both events occurring depends on their relationship:

• If independent: \(\mathrm{P(A \text{ and } B)} = 0.20 \times 0.50 = 0.10 = 10\%\)
• If A is completely contained in B: \(\mathrm{P(A \text{ and } B)} = \mathrm{P(A)} = 20\%\)
• If B is completely contained in A: Impossible since \(\mathrm{P(B)} > \mathrm{P(A)}\)
• If mutually exclusive: \(\mathrm{P(A \text{ and } B)} = 0\%\)

Since we don't know the relationship, \(\mathrm{P(A \text{ and } B)}\) can range from 0% to 20%.

Key Insight for "B or C or Both"

Using the probability addition rule:
\(\mathrm{P(B \text{ or } C)} = \mathrm{P(B)} + \mathrm{P(C)} - \mathrm{P(B \text{ and } C)}\)

The overlap \(\mathrm{P(B \text{ and } C)}\) determines the final value:

• Maximum \(\mathrm{P(B \text{ or } C)}\): When events don't overlap, \(\mathrm{P(B \text{ and } C)} = 0\)
  \(\mathrm{P(B \text{ or } C)} = 50\% + 80\% - 0\% = 130\%\) → But capped at 100%
• Minimum \(\mathrm{P(B \text{ or } C)}\): When overlap is maximum (B completely contained in C)
  \(\mathrm{P(B \text{ and } C)} = \mathrm{P(B)} = 50\%\)
  \(\mathrm{P(B \text{ or } C)} = 50\% + 80\% - 50\% = 80\%\)

Phase 3: Finding the Answer

For "Both A and B"

\(\mathrm{P(A \text{ and } B)}\) must be between 0% and 20%.

Checking answer choices:

• "Must be equal to 40%" ✗ (exceeds maximum of 20%)
• "Must be equal to 50%" ✗ (exceeds maximum of 20%)
• "Must be equal to 70%" ✗ (exceeds maximum of 20%)
• "Must be less than or equal to 20%" ✓ (covers entire possible range)
• "Must be greater than or equal to 80%" ✗ (impossible)

For "B or C or Both"

\(\mathrm{P(B \text{ or } C)}\) must be between 80% and 100%.

Checking answer choices:

• "Must be equal to 40%" ✗ (below minimum of 80%)
• "Must be equal to 50%" ✗ (below minimum of 80%)
• "Must be equal to 70%" ✗ (below minimum of 80%)
• "Must be less than or equal to 20%" ✗ (contradicts minimum of 80%)
• "Must be greater than or equal to 80%" ✓ (covers entire possible range)

Phase 4: Solution

Final Answer:

• Both A and B: Must be less than or equal to 20%
• B or C or Both: Must be greater than or equal to 80%

These selections capture what MUST be true regardless of the specific relationships between the events.