1. Key Concepts and Formulas

• Probability of the Union of Two Events ($P(A \cup B)$): This represents the probability that at least one of the events A or B occurs (i.e., A occurs, or B occurs, or both occur). It is given by the Addition Theorem of Probability: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$
• Probability of Exactly One of Two Events Occurring: This refers to the scenario where event A occurs but event B does not, OR event B occurs but event A does not. It specifically excludes the case where both A and B occur simultaneously. This can be expressed as: $P(\text{exactly one of A, B}) = P(A \cup B) - P(A \cap B)$ This formula is derived by taking the probability of the union (which includes the "both" part) and subtracting the "both" part, leaving only the exclusive occurrences of A or B.
• Probability of the Intersection of Two Events ($P(A \cap B)$): This represents the probability that both event A and event B occur simultaneously. This is the quantity the question asks us to find.

2. Step-by-Step Solution

Step 1: Translate the given information into mathematical expressions.

• What we are doing: We are converting the verbal descriptions of probabilities provided in the problem statement into standard mathematical notation.
• Why we are doing this: This allows us to set up clear equations that can be solved systematically.
  • "the probability that exactly one of them occurs is ${2 \over 5}$": $P(\text{exactly one of A, B}) = {2 \over 5} \quad \text{(Equation 1)}$
  • "the probability that A or B occurs is ${1 \over 2}$": $P(A \cup B) = {1 \over 2} \quad \text{(Equation 2)}$
  • Our objective is to find the value of $P(A \cap B)$.

Step 2: Relate the given probabilities using the appropriate formula.

• What we are doing: We identify the key formula that connects the probability of exactly one event occurring with the probability of their union and intersection.
• Why we are doing this: This formula provides the direct link between the known quantities (from Step 1) and the unknown quantity we need to determine. From our key concepts, the formula for the probability of exactly one of A or B occurring is: $P(\text{exactly one of A, B}) = P(A \cup B) - P(A \cap B) \quad \text{(Equation 3)}$

Step 3: Substitute the known values into the formula and solve for the unknown.

• What we are doing: We substitute the numerical values from Equation 1 and Equation 2 into Equation 3 to form a single equation where $P(A \cap B)$ is the only unknown. Then, we algebraically solve for $P(A \cap B)$.
• Why we are doing this: This is the crucial calculation step that leads directly to our answer. Substitute Equation 1 and Equation 2 into Equation 3: ${2 \over 5} = {1 \over 2} - P(A \cap B)$ Now, rearrange the equation to isolate $P(A \cap B)$: $P(A \cap B) = {1 \over 2} - {2 \over 5}$ To subtract these fractions, we need to find a common denominator. The least common multiple (LCM) of 2 and 5 is 10. Convert the fractions to equivalent fractions with a denominator of 10: $P(A \cap B) = {1 \times 5 \over 2 \times 5} - {2 \times 2 \over 5 \times 2}$ $P(A \cap B) = {5 \over 10} - {4 \over 10}$ Subtract the numerators: $P(A \cap B) = {5 - 4 \over 10}$ $P(A \cap B) = {1 \over 10}$

Step 4: Convert the result to decimal form.

• What we are doing: We convert the fractional probability we obtained into a decimal number.
• Why we are doing this: The given options for the answer are in decimal form, so this conversion allows for easy comparison and selection of the correct option. $P(A \cap B) = 0.10$

3. Common Mistakes & Tips

• Confusing "A or B" with "Exactly One": A frequent mistake is to equate $P(A \cup B)$ (probability of at least one event) with $P(\text{exactly one of A, B})$. Remember that $P(A \cup B)$ includes the case where both A and B occur, whereas "exactly one" explicitly excludes it. The relationship is $P(A \cup B) = P(\text{exactly one of A, B}) + P(A \cap B)$.
• Fraction Arithmetic: Be meticulous when adding or subtracting fractions. Always ensure you find a common denominator before performing the operation to avoid calculation errors.
• Venn Diagram Visualization: For problems involving two or three events, drawing a simple Venn diagram can be extremely helpful. It allows you to visually represent the different regions (A only, B only, A and B, neither A nor B) and understand how the probabilities relate to each other.

4. Summary

This problem tested our understanding of the fundamental relationships between the probabilities of two events. We were given the probability that exactly one of two events occurs and the probability that at least one of them occurs. By applying the key formula $P(\text{exactly one of A, B}) = P(A \cup B) - P(A \cap B)$, we were able to substitute the given values and solve for the probability of both events occurring together, $P(A \cap B)$. The calculation yielded a result of ${1 \over 10}$, which is $0.10$ in decimal form.

5. Final Answer

The probability of both events A and B occurring together is $0.10$. The final answer is \boxed{0.10} which corresponds to option (A).