Problem 6 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Determine whether the events \(A\) and \(B\) are independent. \(P(A)=.6, P(B)=.8, P(A \cap B)=.2\)

Step-by-Step Solution

Answer

Events \(A\) and \(B\) are not independent as \(P(A \cap B) = 0.2\) and \(P(A)P(B) = 0.48\), which are not equal.

Step 1: Write down the given probabilities

We have the following probabilities given: - \(P(A) = 0.6\) - \(P(B) = 0.8\) - \(P(A \cap B) = 0.2\)

Step 2: Calculate the product of \(P(A)\) and \(P(B)\)

Calculate the product of the probabilities \(P(A)\) and \(P(B)\): \[P(A)P(B) = 0.6 \times 0.8 = 0.48\]

Step 3: Compare the product of \(P(A)\) and \(P(B)\) to \(P(A \cap B)\)

Now, we will compare the product of \(P(A)\) and \(P(B)\) to \(P(A \cap B)\). Since \(P(A)P(B) = 0.48\) and \(P(A \cap B) = 0.2\), these values are not equal.

Step 4: Determine if events \(A\) and \(B\) are independent

Since \(P(A \cap B) \neq P(A)P(B)\), events \(A\) and \(B\) are not independent.

Key Concepts

Independent Events

In probability theory, independent events are a fundamental concept. Two events are considered independent if the occurrence of one does not affect the occurrence of another. For example, if you flip a coin and roll a die, the result of the coin flip does not impact the number that comes up on the die.
The criteria for two events, say event \(A\) and event \(B\), to be independent is that the probability of both events occurring is equal to the product of their individual probabilities. This is expressed mathematically as:

• If \( P(A \cap B) = P(A) \times P(B) \), then \(A\) and \(B\) are independent.

Understanding independence helps us evaluate scenarios without one event's occurrence influencing another, thus simplifying probability calculations.

Intersection of Events

The intersection of events refers to the probability that two or more events happen at the same time. In terms of a Venn diagram, it is the overlapping area representing all outcomes common to the events.
The intersection is denoted as \(A \cap B\) for two events \(A\) and \(B\), and it gives us the probability that both events occur simultaneously.

A key aspect of working with intersections is to recognize how it helps define relationships, particularly independence. If you're determining whether events are independent, comparing \(P(A \cap B)\) to \(P(A) \times P(B)\) will tell you if their occurrence is merely a random coincidence or influenced by each other.
This concept is crucial in calculating joint probabilities and understanding the interconnectedness of events.

Calculating Probability

Calculating probability involves determining the likelihood of an event occurring within a defined scenario. It is expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.

Basic probability is calculated using the formula:

• \( P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \)

When dealing with combined events, such as intersections or unions, understanding their definitions and relationships can assist in correct calculations:

• For intersections, use \( P(A \cap B) \).
• For determining probability with independence, compare \( P(A \cap B) \) with \( P(A) \times P(B) \).

Mastering these fundamental techniques will help you solve probability-related problems with confidence by leveraging known probabilities to find unknowns.