Question

Suppose \(P(A)=.1\) and \(P(B)=.5 .\) $$\text { If } P(A \cap B)=0, \text { are } A \text { and } B \text { independent? }$$

Short answer

Answer: No, events A and B are not independent.

Step-by-step solution

Understand the given probabilities

We are given the probabilities of events A and B as: - \(P(A) = 0.1\) - \(P(B) = 0.5\) - \(P(A \cap B) = 0\)

Calculate the product of individual probabilities

Compute the product of the probabilities of events A and B: \(P(A) * P(B) = 0.1 * 0.5 = 0.05\)

Compare the intersection probability and product to determine independence

To verify if events A and B are independent, compare the probability of their intersection, \(P(A \cap B)\), to the product of their individual probabilities, \(P(A) * P(B)\): - \(P(A \cap B) = 0\) - \(P(A) * P(B) = 0.05\) Since \(P(A \cap B) \neq P(A) * P(B)\), events A and B are not independent.

Key concepts

Event Independence

In probability theory, independence is a key concept used to understand the relationship between two events. Two events, say A and B, are independent if the occurrence of one event does not affect the probability of the other occurring.
This implies that learning whether A occurred gives no information about whether B occurred and vice versa. This is a fundamental concept when dealing with probabilities as it simplifies calculating probabilities when events are independent.

• If two events are independent, the probability of both occurring, or their intersection, can be found simply by multiplying the probabilities of each event occurring on its own. This is expressed as: \[ P(A \cap B) = P(A) \times P(B) \]
• It is crucial to differentiate between independence and events being mutually exclusive, which means they cannot occur simultaneously. When events are mutually exclusive, the occurrence of one rules out the occurrence of the other, typically leading to a zero intersection probability. Independence, on the other hand, has nothing to do with mutual exclusivity.

Intersection Probability

Intersection probability helps determine how likely it is for two events to happen at the same time. For instance, if we want to know the probability of both A and B occurring, we refer to the intersection probability, \(P(A \cap B)\). This probability is crucial in determining whether two events are independent.
If the intersection probability aligns with the product of their individual probabilities, then the events are independent. Consequently, any deviation from this equivalence suggests that the events have some form of dependency.
However, if \(P(A \cap B) = 0\), it primarily indicates that the two events cannot occur together – they have no overlap in occurrence. This does not automatically signify independence. Instead, one should check the relationship between \(P(A \cap B)\) and \(P(A) \times P(B)\) to conclude on independence.

Probability Calculation

Probability calculation is a foundational aspect of understanding and predicting event outcomes. It's an essential tool in event analysis whether events are independent or not.

• The first step is specifying the probability of each individual event. In our problem, we have \(P(A) = 0.1\) and \(P(B) = 0.5\).
• When events are evaluated for independence, calculating their joint probability, \(P(A \cap B)\), is vital. In the given problem, this value is 0.
• The comparison between the joint probability and the product of individual probabilities is crucial. For our example, \(P(A \cap B)\) is compared with \(P(A) \times P(B) = 0.05\).

The concept of comparing these values allows us to determine that if \(P(A \cap B)\) differs from the product \(P(A) \times P(B)\), the events are not independent.
This calculation not only helps in verifying independence but is also applicable across various probability scenarios, making it an invaluable skill.