If the probability of an event A is P(A) = 4/9, what are the odds against the occurrence of event A?

Introduction / Context:
This question checks whether you can convert a probability into odds against an event. Many aptitude tests use both probability notation and odds notation, so understanding the relationship between them is essential. Odds against an event compare the chance that the event does not happen with the chance that it does happen.

Given Data / Assumptions:

• Probability of event A occurring, P(A) = 4/9.
• Therefore, probability that event A does not occur, P(A complement) = 1 - 4/9.
• We need odds against event A, written in the form x:y.

Concept / Approach:
For any event A, the relationship between probability and odds is: P(A) = favorable / total. Odds in favor of A are: Odds in favor of A = P(A) : P(A complement). Odds against A are: Odds against A = P(A complement) : P(A). So to find odds against, first compute the probability of A not happening and then express the ratio between this and P(A) in simplest form.

Step-by-Step Solution:
Step 1: Given P(A) = 4/9. Step 2: Compute P(A complement) = 1 - 4/9. Step 3: 1 can be written as 9/9, so P(A complement) = 9/9 - 4/9 = 5/9. Step 4: Odds against A are given by P(A complement) : P(A). Step 5: So the raw ratio is (5/9) : (4/9). Step 6: Multiply both sides of the ratio by 9 to remove denominators, giving 5 : 4. Step 7: Therefore, the odds against event A are 5:4.

Verification / Alternative check:
Imagine there are 9 equally likely outcomes. For 4 of them event A occurs, and for 5 of them it does not occur. Odds against A compare the count of failures to the count of successes, which would be 5 unfavorable outcomes and 4 favorable outcomes. This directly produces odds against A equal to 5:4, matching the calculation using formal probability complements.

Why Other Options Are Wrong:
Option 4:5 represents odds in favor of A, not odds against A. Option 4:9 incorrectly uses the original probability with total outcomes; it is not a comparison of success and failure. Option 9:4 would correspond to a situation where the event is more likely to fail than succeed by a larger margin, which does not match P(A) = 4/9. Option 1:2 is unrelated to the given probability and does not come from any correct calculation based on P(A) = 4/9.

Common Pitfalls:
Students often confuse odds with probabilities. Odds are always a ratio of two counts or probabilities, not a single fraction between 0 and 1. Another mistake is reversing odds in favor and odds against. Always remember: odds against A compare failure to success (not success to failure). Also, do not forget to simplify ratios into the smallest whole number form so that they are easy to interpret in exam conditions.

Final Answer:
The odds against the occurrence of event A are 5:4.