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Chapter 2: Problem 143

Suppose that \(P(A \mid B)=0.4, \quad P\left(A \mid B^{\prime}\right)=0.2\), and \(P(B)=0.8 .\) Determine \(P(B \mid A) .\)

Short Answer

Expert verified
The probability \( P(B \mid A) \) is \( \frac{8}{9} \).

Step by step solution

01

Understanding the Given Information

We're given the conditional probabilities \( P(A \mid B) = 0.4 \) and \( P(A \mid B') = 0.2 \) along with the probability \( P(B) = 0.8 \). Our goal is to find \( P(B \mid A) \).
02

Using Bayes' Theorem

Bayes' Theorem relates conditional probabilities and can be expressed as:\[P(B \mid A) = \frac{P(A \mid B) P(B)}{P(A)}\]To solve for \( P(B \mid A) \), we need to find \( P(A) \).
03

Finding P(A) using the Law of Total Probability

The Law of Total Probability says:\[P(A) = P(A \mid B) P(B) + P(A \mid B') P(B')\]Substituting the given values and noting that \( P(B') = 1 - P(B) = 0.2 \), we find:\[P(A) = 0.4 \times 0.8 + 0.2 \times 0.2 = 0.32 + 0.04 = 0.36\]
04

Calculating P(B | A)

Substitute \( P(A) = 0.36 \), \( P(A \mid B) = 0.4 \), and \( P(B) = 0.8 \) into Bayes' Theorem:\[P(B \mid A) = \frac{0.4 \times 0.8}{0.36} = \frac{0.32}{0.36} = \frac{8}{9}\]
05

Conclusion

The probability \( P(B \mid A) \) is \( \frac{8}{9} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Conditional Probability
Conditional probability is a way of expressing the probability of an event occurring, given that another event has already occurred. It answers the question "If we know event B has happened, what are the chances that event A will happen?"

Mathematically, this is represented by the formula:
  • \( P(A \mid B) = \frac{P(A \cap B)}{P(B)} \)
Here鈥檚 what the notation means:
  • \( P(A \mid B) \) is the probability of A given B.
  • \( P(A \cap B) \) is the probability that both A and B occur.
  • \( P(B) \) is the probability of B.
Conditional probability allows us to adjust our perception of how likely an event is, based on new information. It鈥檚 a key concept in understanding how one event can affect the likelihood of another.
Law of Total Probability
The Law of Total Probability is a fundamental rule for calculating probabilities that are broken into subsets. It helps us find the probability of a total event by considering all possible scenarios that lead to this event. This law is especially useful when you can split an event into mutually exclusive and exhaustive events.

The law is expressed as:
  • \( P(A) = P(A \mid B)P(B) + P(A \mid B')P(B') \)
In the formula above:
  • \( P(A \mid B)P(B) \) is the probability of A occurring with B.
  • \( P(A \mid B')P(B') \) is the probability of A occurring without B.
  • \( B' \) represents "not B", also known as the complement of B.
By applying the Law of Total Probability, you consider both situations: where B happens and where it doesn鈥檛. Summing these values provides the total probability of A.
Probability Theory
Probability theory is the branch of mathematics that deals with the analysis of random events. It鈥檚 centered around understanding how likely an event is to occur.

Basic ideas include:
  • An event is any outcome or set of outcomes from some random process.
  • The probability of an event is a numerical measure of the likelihood that the event will occur.
Probabilities are calculated following these rules:
  • All probabilities are numbers between 0 and 1, inclusive.
  • If an event is certain to happen, its probability is 1.
  • If an event is certain not to happen, its probability is 0.
  • The sum of probabilities of all possible outcomes of an event equals 1.
Probabilities can be combined and calculated using concepts such as conditional probability, the Law of Total Probability, and Bayes' Theorem. These all help in making sense of uncertain situations and guide decision-making under uncertainty.

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Most popular questions from this chapter

In the design of an electromechanical product, 12 components are to be stacked into a cylindrical casing in a manner that minimizes the impact of shocks. One end of the casing is designated as the bottom and the other end is the top. (a) If all components are different, how many different designs are possible? (b) If seven components are identical to one another, but the others are different, how many different designs are possible? (c) If three components are of one type and identical to one another, and four components are of another type and identical to one another, but the others are different, how many different designs are possible?

A manufacturing process consists of 10 operations that can be completed in any order. How many different production sequences are possible?

An inspector working for a manufacturing company has a \(99 \%\) chance of correctly identifying defective items and a \(0.5 \%\) chance of incorrectly classifying a good item as defective. The company has evidence that its line produces \(0.9 \%\) of nonconforming items. (a) What is the probability that an item selected for inspection is classified as defective? (b) If an item selected at random is classified as nondefective, what is the probability that it is indeed good?

Each of the possible five outcomes of a random experiment is equally likely. The sample space is \(\\{a, b, c, d, e\\} .\) Let \(A\) denote the event \(\\{a, b\\},\) and let \(B\) denote the event \(\\{c, d, e\\} .\) Determine the following (a) \(P(A)\) (b) \(P(B)\) (c) \(P\left(A^{\prime}\right)\) (d) \(P(A \cup B)\) (e) \(P(A \cap B)\)

In a chemical plant, 24 holding tanks are used for final product storage. Four tanks are selected at random and without replacement. Suppose that six of the tanks contain material in which the viscosity exceeds the customer requirements. (a) What is the probability that exactly one tank in the sample contains high- viscosity material? (b) What is the probability that at least one tank in the sample contains high-viscosity material? (c) In addition to the six tanks with high viscosity levels, four different tanks contain material with high impurities. What is the probability that exactly one tank in the sample contains high-viscosity material and exactly one tank in the sample contains material with high impurities?
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