Events K and L are defined on the same sample space. Given that P(K) = 0.8 and P(L) = 0.6, can K and L be disjoint events?

Introduction / Context:
This question tests conceptual understanding of disjoint (mutually exclusive) events in probability theory. Events are called disjoint if they cannot occur at the same time, which has a direct implication for the relationship between their probabilities. We are given P(K) and P(L) and asked whether K and L can be disjoint, so we must recall the basic rule for mutually exclusive events.

Given Data / Assumptions:

• P(K) = 0.8.
• P(L) = 0.6.
• K and L are events on the same sample space.
• All probabilities must lie between 0 and 1.

Concept / Approach:
If two events K and L are disjoint, then P(K and L) = 0, and the probability of their union satisfies P(K or L) = P(K) + P(L). Since probabilities of any event cannot exceed 1, this sum must be less than or equal to 1 for disjoint events. Therefore, to check if K and L can be disjoint, we simply add their probabilities and see whether the result is at most 1.

Step-by-Step Solution:
For disjoint events, the rule is P(K or L) = P(K) + P(L). Given P(K) = 0.8 and P(L) = 0.6. Compute P(K) + P(L) = 0.8 + 0.6 = 1.4. For any event in a probability space, the probability cannot exceed 1. If K and L were disjoint, we would have P(K or L) = 1.4, which is impossible. Therefore, K and L cannot be disjoint events.

Verification / Alternative check:
Another way to argue is to note that for any two events, P(K or L) ≤ 1. When events are disjoint, P(K or L) equals the sum P(K) + P(L). Therefore, for disjoint events, P(K) + P(L) must be less than or equal to 1. Here, the sum is 1.4, which violates this basic requirement. So we can be completely certain that K and L cannot be mutually exclusive and must have some common outcomes with positive probability.

Why Other Options Are Wrong:
Yes, they can be disjoint events: This ignores the fact that the sum of probabilities exceeds 1, which is impossible for disjoint events.
They are always independent: The information given does not guarantee independence, and in fact, the large probabilities plus non disjoint condition say nothing about independence.
There is not enough information to decide: There is enough information because the numerical values directly show that the disjoint condition cannot hold.
Both events must be complements: Complementary events have probabilities that sum to exactly 1, not 1.4.

Common Pitfalls:
A frequent confusion is between independence and disjointness. Disjoint events cannot happen together, while independent events may occur together but do not influence each other in terms of probability. Another common mistake is forgetting that probabilities cannot exceed 1, which is crucial here. Always check the sum of probabilities when dealing with mutually exclusive events.

Final Answer:
Hence, K and L cannot be disjoint events. The correct statement is No, they cannot be disjoint events.