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Chapter 4: Problem 51

Given that \(P(A \mid B)=.44\) and \(P(A\) and \(B)=.33\), find \(P(B)\).

Short Answer

Expert verified
The probability of event B occurring is \(0.75\).

Step by step solution

01

Understanding the formula

First, we need to understand the formula of conditional probability which is given by: \(P(A and B) = P(A | B) * P(B)\). This formula expresses the relationship between the probability of events A and B happening together and the probability of event A happening given that B has happened.
02

Rearranging the formula

From the formula above, we can express the probability of event B happening in terms of the other known quantities: \(P(B) = P(A and B) / P(A | B)\). The solution comes straight from this formula.
03

Substituting the values into the formula

Now we substitute the given values into the formula: \(P(B) = .33 / .44\). If we divide these two values, we get the probability of event B occurring.

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Key Concepts

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

Probability Theory
Probability theory is the mathematical framework that allows us to reason about uncertainty and randomness. It helps in predicting how likely events are to happen. This field uses numbers between 0 and 1 to express the probability of an event, with 0 indicating it鈥檚 impossible and 1 meaning it鈥檚 certain.

Here's a quick overview:
  • Probability can describe a single event occurring, such as rolling a die and getting a six, or multiple events, like rolling two dice and both showing sixes.
  • The total probability of all possible outcomes of a random process always equals 1.
  • Cross connections exist between events, often described by concepts like independent, mutually exclusive, or conditional probabilities.
Understanding these basics can help us tackle various types of problems, whether simple or complex.
Bayes' Theorem
Bayes' Theorem is a powerful concept in probability theory. It provides a way to update our beliefs about the likelihood of an event based on new evidence. This theorem is particularly useful when we deal with conditional probabilities.

Bayes' Theorem states that:\[P(A | B) = \frac{P(B | A) \cdot P(A)}{P(B)}\]This equation helps us understand how the probability of event A occurring changes if we know that event B has occurred. Typically, it relates "new" and "prior" information.
  • P(A | B): Probability of A given B.
  • P(B | A): Probability of B given A.
  • P(A): Overall probability of A occurring.
  • P(B): Overall probability of B occurring.
Using Bayes' Theorem, we can approach problems from a perspective of evidence keeping, where data updates our understanding of events.
Probability of Events
The probability of events refers to the chance that particular outcomes will occur in a random experiment. We often deal with combinations of events, which is where joint and conditional probabilities come into play.

Some important points about the probability of events include:
  • Joint Probability: This is the probability that two events happen at the same time, expressed as \(P(A \text{ and } B)\).
  • Conditional Probability: This measures how likely an event A is to happen given that another event B has already occurred. It's written as \(P(A | B)\).
  • The formula \(P(A \text{ and } B) = P(A | B) \cdot P(B)\) helps in finding the occurrence of event B using known quantities.
Understanding these concepts facilitates breaking down problems into manageable parts, like solving for unknown probabilities as seen in many textbook exercises.

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Most popular questions from this chapter

In a group of adults, some own iPads, and others do not. If two adults are randomly selected from this group, how many total outcomes are possible? Draw a tree diagram for this experiment.

A thief has stolen Roger's automatic teller machine (ATM) card. The card has a four-digit personal identification number (PIN). The thief knows that the first two digits are 3 and 5 , but he does not know the last two digits. Thus, the PIN could be any number from 3500 to \(3599 .\) To protect the customer, the automatic teller machine will not allow more than three unsuccessful attempts to enter the PIN. After the third wrong PIN, the machine keeps the card and allows no further attempts. a. What is the probability that the thief will find the correct PIN within three tries? (Assume that the thief will not try the same wrong PIN twice.) b. If the thief knew that the first two digits were 3 and 5 and that the third digit was either 1 or 7 , what is the probability of the thief guessing the correct PIN in three attempts?

In a group of adults, some own iPads, and others do not. Two adults are randomly selected from this group. List all the outcomes included in each of the following events. Indicate which are simple and which are compound events. a. One person owns an iPad and the other does not. b. At least one person owns an iPad. c. Not more than one person owns an iPad. d. The first person owns an iPad and the second does not.

A random sample of 2000 adults showed that 1320 of them have shopped at least once on the Internet. What is the (approximate) probability that a randomly selected adult has shopped on the Internet?

Jane and Mike are planning to go on a two-week vacation next summer. They have selected six vacation resorts, two of which are in Canada and remaining four are in Caribbean countries. Jane prefers going to a Canadian resort, and Mike prefers to vacation in one of the Caribbean countries. After much argument, they decide that they will put six balls of the same size, each marked with one of the six vacation resorts, in a hat. Then they will ask their 8 -year- old son to randomly choose one ball from these six balls. What is the probability that a vacation resort from the Caribbean countries is selected? Is this an example of the classical approach, relative frequency approach, or the subjective probability approach? Explain your answer. Do these probabilities add to \(1.0 ?\) If yes, why?
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