The probability that events E and F will both occur is 0.42. The probability that event E will occur is 0.58.
Column A: The probability that event F will occur
Column B: 0.58
Column A: The probability that event F will occur
Column B: 0.58
Show Hint
- Quantity A is greater
- Quantity B is greater
- The two quantities are equal
- The relationship cannot be determined from the information given
The Correct Option is D
Solution and Explanation
Step 1: Understanding the Concept:
This question deals with the relationship between the probabilities of individual events and their joint probability. We need to determine the possible range for the probability of event F, P(F).
Step 2: Key Formula or Approach:
We will use two fundamental rules of probability:
1. The probability of the intersection of two events is always less than or equal to the probability of each individual event: \(P(E \text{ and } F) \leq P(F)\).
2. The probability of the union of two events cannot exceed 1: \(P(E \text{ or } F) = P(E) + P(F) - P(E \text{ and } F) \leq 1\).
Step 3: Detailed Explanation:
Finding the lower bound for P(F):
From rule 1, we know that \(P(F)\) must be at least as large as the probability of both E and F occurring.
\[ P(F) \geq P(E \text{ and } F) \] \[ P(F) \geq 0.42 \] Finding the upper bound for P(F):
From rule 2, we use the formula for the probability of the union of two events.
\[ P(E) + P(F) - P(E \text{ and } F) \leq 1 \] Substitute the given values:
\[ 0.58 + P(F) - 0.42 \leq 1 \] \[ 0.16 + P(F) \leq 1 \] \[ P(F) \leq 1 - 0.16 \] \[ P(F) \leq 0.84 \] So, the possible range for P(F) is \(0.42 \leq P(F) \leq 0.84\).
Step 4: Comparing the Quantities:
We are comparing Column A, P(F), with Column B, 0.58.
The value of P(F) can be anywhere between 0.42 and 0.84. This range includes values less than, equal to, and greater than 0.58.
- For example, P(F) could be 0.50 (less than 0.58).
- P(F) could be 0.58 (equal to 0.58).
- P(F) could be 0.70 (greater than 0.58).
Since we cannot determine a fixed relationship, the answer is (D).
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