Problem 3 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Let \(A\) and \(B\) be two events in a sample space \(S\) such that \(P(A)=.6\) and \(P(B \mid A)=.5\). Find \(P(A \cap B)\).

Step-by-Step Solution

Answer

To find the joint probability \(P(A \cap B)\), we use the formula \(P(A \cap B) = P(B \mid A) \times P(A)\). Given that \(P(A) = 0.6\) and \(P(B \mid A) = 0.5\), we calculate \(P(A \cap B) = 0.5 \times 0.6 = 0.3\).

Step 1: Write down the given probabilities

We have the following probabilities given in the exercise: $$P(A) = 0.6$$ $$P(B \mid A) = 0.5$$

Step 2: Rearrange the formula for conditional probability to find the joint probability

We need to find the joint probability \(P(A \cap B)\). The formula for conditional probability is: $$P(B \mid A) = \dfrac{P(A \cap B)}{P(A)}$$ Rearrange the formula to isolate \(P(A \cap B)\): $$P(A \cap B) = P(B \mid A) \times P(A)$$

Step 3: Compute the joint probability using the given values

Now we substitute the given probabilities into the rearranged formula: $$P(A \cap B) = 0.5 \times 0.6$$ Calculate the result: $$P(A \cap B) = 0.3$$ So, the probability of the intersection of events A and B, \(P(A \cap B)\), is 0.3.

Key Concepts

Conditional Probability

Conditional probability is essential when you want to determine the likelihood of an event occurring, given that another event has already taken place. Let's break it down a bit more. Suppose there are two events, say, event A and event B, within a sample space. The "conditional" part refers to the fact that we are not merely interested in event B happening, but in it happening after we already know that event A has happened.

• Mathematically, it's expressed as \( P(B \mid A) \), meaning "the probability of B given A".
• This probability helps in understanding dependencies between events, which is critical in many real-world scenarios, like weather forecasting or medical diagnosis.

How do we calculate conditional probability? Use the formula:
\[ P(B \mid A) = \frac{P(A \cap B)}{P(A)} \]
This equation tells us how probable B is, after assuming A has already happened. It's always calculated by considering the occurrence of a particular condition, thus ensuring a more precise probability estimate.

Joint Probability

Joint probability is the likelihood of two events happening at the same time, or simultaneously. If you're interested in knowing the probability that both events A and B occur, you will look for their joint probability.

• It's quantified as \( P(A \cap B) \), meaning "the probability of A and B".
• In practical terms, it could refer to scenarios like picking a card that is both red and a king from a deck of playing cards.

Joint probability is closely related to conditional probability. In fact, you can determine joint probability when conditional probability is known, using the formula:
\[ P(A \cap B) = P(B \mid A) \times P(A) \]
This is exactly what the step-by-step solution does: it uses the given conditional probability \( P(B \mid A) \) and the probability of A to find the joint probability \( P(A \cap B) \). Understanding joint probability helps in situations where we deal with cumulative events, puzzles, and even certain business decisions.

Sample Space

The concept of the sample space is the foundational block of probability theory. A sample space, often denoted by S, is the set of all possible outcomes of a particular experiment or random trial.

• For a coin toss, the sample space is \( \{ \text{Heads}, \text{Tails} \} \).
• Rolling a six-sided die has a sample space \( \{ 1, 2, 3, 4, 5, 6 \} \).

In our problem, the sample space, S, would include all possible combinations of events A and B, among others.
Understanding the sample space is crucial because it sets the stage for defining events and calculating their probabilities. To define event probabilities accurately, you must first grasp what outcomes are possible. When dealing with complex probability problems like those with conditional or joint probabilities, identifying and understanding the sample space make the task far more manageable.
Hence, building an intuitively correct sample space is a vital skill when tackling probabilistic scenarios, laying the groundwork for more advanced topics and calculations.