Problem 122 - Applied Statistics and Probability for Engineers

Question

Suppose that \(P(A \mid B)=0.2, P\left(A \mid B^{\prime}\right)=0.3,\) and \(P(B)=0.8 .\) What is \(P(A) ?\)

Step-by-Step Solution

Answer

P(A) = 0.22

Step 1: Understanding Conditional Probability

We have two conditional probabilities given: \(P(A \mid B)\) which is the probability of event \(A\) occurring given that \(B\) has occurred, and \(P(A \mid B')\) which is the probability of \(A\) occurring given that \(B\) has not occurred. We also have \(P(B)\), the probability of \(B\). Our goal is to find the unconditional probability \(P(A)\).

Step 2: Using the Law of Total Probability

The law of total probability tells us how to evaluate the probability of \(A\) in relation to \(B\) and \(B'\). It is given by the formula:\[ P(A) = P(A \cap B) + P(A \cap B') \]Using the definition of conditional probability, \(P(A \cap B) = P(A \mid B) P(B)\) and \(P(A \cap B') = P(A \mid B') P(B')\).

Step 3: Calculate \(P(A \cap B)\)

Apply the definition of conditional probability:\[ P(A \cap B) = P(A \mid B) P(B) = 0.2 \times 0.8 = 0.16 \]

Step 4: Calculate \(P(A \cap B')\)

First, find \(P(B')\) which is the complement of \(P(B)\):\[ P(B') = 1 - P(B) = 0.2 \]Now apply the definition of conditional probability:\[ P(A \cap B') = P(A \mid B') P(B') = 0.3 \times 0.2 = 0.06 \]

Step 5: Compute \(P(A)\)

Now use the values obtained to find \(P(A)\):\[ P(A) = P(A \cap B) + P(A \cap B') = 0.16 + 0.06 = 0.22 \]

Key Concepts

Law of Total Probability

The Law of Total Probability is a fundamental principle in probability theory that helps us manage complex scenarios by breaking them down into simpler parts. This rule is particularly useful when you're dealing with conditional probabilities and you want to find the overall probability of an event.

If you have partitions of your sample space, this law allows you to evaluate the overall probability of an event by considering all possible ways the event can occur. In our exercise, the probability of event \(A\) is divided over two mutually exclusive scenarios: when \(B\) occurs and when \(B'\) (not \(B\)) occurs.

The formula for the Law of Total Probability is:

• \[ P(A) = P(A \cap B) + P(A \cap B') \]

Using conditional probabilities, this becomes:

• \[ P(A) = P(A \mid B) P(B) + P(A \mid B') P(B') \]

This law allows us to compute the probability of complex events by summing the probabilities from each portion of the sample space.

Unconditional Probability

Unconditional probability refers to the probability of an event occurring without any precondition or restriction. It's also known as marginal probability.

In our example, we're interested in finding the unconditional probability \(P(A)\), which represents the likelihood of event \(A\) occurring regardless of whether \(B\) happens or not.

Unconditional probability is foundational because it provides a comprehensive view of an event's likelihood in the absence of any other condition.

To find it, we used previously determined conditional probabilities \(P(A \mid B)\) and \(P(A \mid B')\) and the probability of \(B\), using the Law of Total Probability to piece together the complete picture:

• \[ P(A) = 0.16 + 0.06 = 0.22 \]

Here, we added the probabilities from each scenario (\(B\) and \(B'\)) to get the unconditional probability.

Probability of Events

Probability of events refers to the likelihood or chance of an event happening. Probability evaluates how likely it is for a given event to occur out of all possible outcomes. Each event in a probability space has an associated probability value between 0 and 1, where 0 means the event cannot happen, and 1 means it certainly will happen.

The exercise we're discussing involves conditional probabilities as well as their use to derive other types of probabilities. The example given uses information about the event \(A\) with respect to another event \(B\) and its complement \(B'\).

Viewing probability through the lens of events:

• Conditional Probability: Likelihood of an event given another event has occurred (e.g., \(P(A \mid B)\)).
• Unconditional Probability: Overall likelihood of an event (e.g., \(P(A)\)).

Understanding how to compute these different probabilities empowers you to see how events happen and interact within a given sample space.