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Chapter 4: Problem 82

Suppose that \(\mathrm{A}\) and \(\mathrm{B}\) are events defined on a common sample space and that the following probabilities are known: \(P(A \text { or } B)=0.7, P(B)=0.5,\) and \(P(A | B)=0.2\) Find \(P(\mathrm{A})\)

Short Answer

Expert verified
The probability of event A, P(A), is 0.3.

Step by step solution

01

Calculate P(A and B)

First, find the probability of both A and B occurring. This can be computed using the conditional probability. The formula for conditional probability is P(A|B) = P(A and B) / P(B). Given P(A|B) = 0.2 and P(B) = 0.5, we can solve the equation for P(A and B) to get: P(A and B) = P(A|B) * P(B) = 0.2 * 0.5 = 0.1.
02

Apply the 'OR' Rule of Probabilities

Next, utilize the 'OR' rule of probabilities, which states that P(A or B) = P(A) + P(B) - P(A and B). We know that P(A or B) = 0.7, P(B) = 0.5, and from step 1, P(A and B) = 0.1. Substituting these values into the 'OR' rule equation, we find: 0.7 = P(A) + 0.5 - 0.1.
03

Solve for P(A)

Lastly, solve the equation from step 2 to find P(A). We rearrange the equation to get: P(A) = 0.7 - 0.5 + 0.1 = 0.3. Hence, the probability of event A is 0.3.

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Key Concepts

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

Probability Theory
Probability theory helps us understand the likelihood of different outcomes. It involves calculating chances for events to occur. In this context, probability ranges between 0 and 1.
Understanding probability is a crucial skill in various fields, such as finance, engineering, and science.
Key concepts include:
  • **Random Experiment**: An action with unpredictable outcomes, like rolling a die.
  • **Outcome**: A possible result of the random experiment.
  • **Event**: A specific set of outcomes we are interested in. For example, rolling an even number on a die.
The probability of an event provides insight into how likely it is to happen.
In our exercise, events A and B have probabilities that help us calculate unknown probabilities using known formulas.
Sample Space
The sample space is the set of all possible outcomes of a random experiment.
It's like a list of everything that could happen when you perform an experiment.
For instance, if you flip a coin, the sample space is \(\{ \ ext{Heads, Tails} \}\).
Understanding the sample space is fundamental because:
  • It provides the framework for determining probabilities of events.
  • It allows us to ensure all potential outcomes are considered when calculating probabilities.
In our problem, we analyze how events \(A\) and \(B\) interact within the sample space.
Considering "or" and "and" scenarios helps us navigate through these probabilities accurately.
Event Probability Calculations
Event probability calculations involve determining the likelihood of one or more events occurring.
Some important rules and formulas help in these calculations:
  • **Conditional Probability**: \(P(A|B) = \frac{P(A \text{ and } B)}{P(B)}\), helps find the probability of \(A\) given \(B\) has occurred.
  • **Addition Rule**: \(P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)\), calculates the probability of either event \(A\) or \(B\) happening.
By applying these rules, you can solve more complex problems.
In our original problem, these rules allow us to find \(P(A)\), given \(P(A \text{ or } B)\), \(P(B)\), and \(P(A|B)\).
Careful application of these formulas leads to accurate results and a deeper understanding of how probabilities work.

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Most popular questions from this chapter

Two flower seeds are randomly selected from a package that contains five seeds for red flowers and three seeds for white flowers. a. What is the probability that both seeds will result in red flowers? b. What is the probability that one of each color is selected? c. What is the probability that both seeds are for white flowers?

Suppose a box of marbles contains equal numbers of red marbles and yellow marbles but twice as many green marbles as red marbles. Draw one marble from the box and observe its color. Assign probabilities to the elements in the sample space.

Simulates generating a family. The "family" will stop having children when it has a boy or three girls, whichever comes first. Assuming that a woman is equally likely to bear a boy or a girl, perform the simulation 24 times. What is the probability that the family will have a boy?

Events \(\mathrm{R}\) and \(\mathrm{S}\) are defined on a sample space. If \(P(R)=0.2\) and \(P(S)=0.5,\) explain why each of the following statements is either true or false: a. If \(\mathrm{R}\) and \(\mathrm{S}\) are mutually exclusive, then \(P(\mathrm{R} \text { or } \mathrm{S})=0.10\) b. If \(R\) and \(S\) are independent, then \(P(R \text { or } S)=0.6\) c. If \(R\) and \(S\) are mutually exclusive, then \(P(R \text { and } S)=0.7\) d. If \(R\) and \(S\) are mutually exclusive, then \(P(\mathrm{R} \text { or } \mathrm{S})=0.6\)

A single card is drawn from a standard deck. Let A be the event that "the card is a face card" (a jack, a queen, or a king), \(\mathbf{B}\) is a "red card," and \(\mathrm{C}\) is "the card is a heart." Determine whether the following pairs of events are independent or dependent: a. \(\quad A\) and \(B\) b. \(\mathrm{A}\) and \(\mathrm{C}\) c. \(\quad B\) and \(C\)
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