For two events A and B, the probability of event A is 1/4, the probability of event B is 1/2, and the probability that both A and B occur together is 7/50. What is the probability that neither event A nor event B occurs?

Introduction / Context:
This problem uses the basic rules of probability for two events, including the formula for the probability of a union and its complement. You are given probabilities of events A, B, and their intersection, and you must compute the probability that neither event occurs. This is standard in quantitative aptitude and helps reinforce the relationship between unions, intersections, and complements.

Given Data / Assumptions:

• P(A) = 1/4.
• P(B) = 1/2.
• P(A and B) = 7/50.
• We need P(neither A nor B), which is the probability that event A does not occur and event B does not occur.

Concept / Approach:
First, find the probability that at least one of A or B occurs, which is P(A or B). For any two events: P(A or B) = P(A) + P(B) - P(A and B). Then use the complement rule. The event neither A nor B is the complement of A or B. Thus: P(neither A nor B) = 1 - P(A or B).

Step-by-Step Solution:
Step 1: Write down the formula for P(A or B). P(A or B) = P(A) + P(B) - P(A and B). Step 2: Substitute the given values: P(A) = 1/4, P(B) = 1/2, P(A and B) = 7/50. Step 3: Convert all fractions to a common denominator, for example 100. 1/4 = 25/100, 1/2 = 50/100, and 7/50 = 14/100. Step 4: Compute P(A or B) using these equivalent fractions. P(A or B) = 25/100 + 50/100 - 14/100 = (25 + 50 - 14) / 100 = 61/100. Step 5: Use the complement rule to find P(neither A nor B). P(neither A nor B) = 1 - P(A or B) = 1 - 61/100. Step 6: Convert 1 to 100/100 and subtract to get 100/100 - 61/100 = 39/100.

Verification / Alternative check:
You can interpret the probabilities as percentages of a large population or of many repeated trials. For example, imagine 100 trials. On average, A occurs in 25 trials, B occurs in 50 trials, and both occur together in 14 trials. By inclusion exclusion, A or B occurs in 25 + 50 - 14 = 61 trials. Therefore, in the remaining 39 trials, neither A nor B occurs, giving a proportion of 39/100. This reasoning directly matches the computed probability.

Why Other Options Are Wrong:
Option 61/100 is actually P(A or B), not the probability of neither event occurring. Option 17/100 and option 25/100 are smaller than the correct complement and would arise from subtracting incorrectly or misusing the intersection term. Option 49/100 does not align with either 61/100 or its complement and does not come from a consistent application of the union formula and complement rule. Only 39/100 fits the given information correctly.

Common Pitfalls:
Students sometimes forget to subtract P(A and B) when computing P(A or B), which double counts the overlap and leads to a union probability greater than 1 in extreme cases. Another mistake is to confuse P(neither A nor B) with P(A complement) * P(B complement), which is only valid if A and B are independent and P(A and B) equals P(A) * P(B). Because we are explicitly given P(A and B), we should use that data instead of assuming independence. Always follow the union formula and complement rule step by step to avoid these errors.

Final Answer:
The probability that neither A nor B occurs is 39/100.