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Chapter 4: Problem 74

A and B are events defined on a sample space, with \(P(\mathrm{B})=0.5\) and \(P(\mathrm{A} \text { and } \mathrm{B})=0.4 .\) Find \(P(\mathrm{A} | \mathrm{B})\)

Short Answer

Expert verified
The conditional probability P(A|B) is 0.8.

Step by step solution

01

Identify Known Values

Identify known quantities from the problem statement. The probability of B, \( P(B) = 0.5 \), and the probability of A and B, \( P(A \text{ and } B) = 0.4 \).
02

Apply Conditional Probability Formula

Substitute these known values into the formula for conditional probability: \[ P(A|B) = \frac{P(A \text{ and } B)}{P(B)} \]. Therefore, \[ P(A|B) = \frac{0.4}{0.5} \].
03

Solve the Equation

Calculate the quotient to find \( P(A|B) \).

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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 a branch of mathematics that deals with the study of randomness and uncertainty. It provides the framework for quantifying the likelihood of various outcomes in a random experiment. Probability allows us to predict how likely an event is to occur.
These predictions are made within a given sample space, which includes all possible outcomes. By assigning numeric values between 0 and 1, where 0 means the event cannot happen and 1 means the event is certain to happen, we can model real-world uncertainties. This forms the foundation for more complex concepts, such as conditional probability.
Events in Probability
In probability theory, an event is any collection of outcomes from a sample space. Events can be defined to encompass single outcomes or groups of outcomes. Understanding events is crucial since probability aims at determining the likelihood of these events.
For example, imagine rolling a dice. Each face of the dice is an outcome, and rolling an even number is an event that includes outcomes: 2, 4, and 6. Events can often be combined using operations like "and" (intersection) and "or" (union), which help in constructing compound events from simpler ones.
Sample Space
The sample space in a probability experiment is the set of all possible outcomes. It provides a complete list that comprehensively captures every potential result of an experiment.
For instance, in a coin toss, the sample space includes two outcomes: heads and tails. Defining the sample space correctly is crucial since it sets the stage for accurately determining probabilities for different events associated with the experiment.
  • Discrete sample spaces involve a finite set of outcomes, such as the roll of a die.
  • Continuous sample spaces contain infinitely many possible outcomes, like the measurement of rainfall.
Probability Formulas
Probability formulas are specific mathematical expressions used to calculate the likelihood of various events. One of the most common formulas is conditional probability, which assesses the probability of an event occurring, given that another event has already occurred.
  • Conditional Probability Formula: The formula is expressed as \( P(A|B) = \frac{P(A \text{ and } B)}{P(B)} \). This calculates the probability of event A occurring, provided that event B has already happened.
  • Intersection: The term \( P(A \text{ and } B) \) refers to the joint probability of events A and B happening together.
  • Divisor: \( P(B) \) is the probability of the conditioning event, in this case, B.

These formulas simplify the process of understanding complex dependencies between different events, such as in the provided exercise where we calculated \( P(A|B) \) using these principles.

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Most popular questions from this chapter

\(\mathrm{A}\) chocolate kiss is to be tossed into the air and will be landing on a smooth hard surface (similar to tossing a coin or rolling dice). a. What proportion of the time do you believe the kiss will land "point up" \(\bigoplus\) (as opposed to "point \(\left.\operatorname{down}^{\prime \prime}()\right) ?\) b. Let's estimate the probability that a chocolate kiss lands "point up" when it lands on a smooth hard surface after being tossed. Using a chocolate kiss, with the wrapper still on, perform the die experiment discussed on pages \(180-181 .\) Toss the kiss 10 times, record the number of "point up" landings (or put 10 kisses in a cup, shake and dump them onto a hard smooth surface, and use each toss for a block of 10 ), and record the results. Repeat until you have 200 tosses. Chart and graph the data as individual sets of 10 and as cumulative relative frequencies. c. What is your best estimate for the true \(P(\bigotimes) ?\) Explain. d. If unwrapped kisses were to be tossed, what do you think the probability of "point up" landings would be? Would it be different? Explain. e. Unwrap the chocolate kisses used in part b and repeat the experiment. f. Are the results in part e what you anticipated? Explain.

Explain why \(P(\text { A occurring when } \mathrm{B} \text { has occurred) }=0\) when events \(\mathrm{A}\) and \(\mathrm{B}\) are mutually exclusive.

Explain why these probabilities cannot be legitimate: \(P(\mathrm{A})=0.6, P(\mathrm{B})=0.4, P(\mathrm{A} \text { and } \mathrm{B})=0.7\)

A company that manufactures shoes has three factories. Factory 1 produces \(25 \%\) of the company's shoes, Factory 2 produces \(60 \%,\) and Factory 3 produces \(15 \%\) One percent of the shoes produced by Factory 1 are mislabeled, \(0.5 \%\) of those produced by Factory 2 are mislabeled, and \(2 \%\) of those produced by Factory 3 are mislabeled. If you purchase one pair of shoes manufactured by this company, what is the probability that the shoes are mislabeled?

Determine whether each of the following pairs of events is independent: a. Rolling a pair of dice and observing a " 2 " on one of the dice and having a "total of \(10 "\) b. Drawing one card from a regular deck of playing cards and having a "red" card and having an "ace" c. Raining today and passing today's exam d. Raining today and playing golf today e. Completing today's homework assignment and being on time for class
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