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Chapter 5: Problem 9

Suppose events \(E\) and \(F\) are independent, \(P(E)=0.3\), and \(P(F)=0.6 .\) What is the \(P(E \text { and } F) ?\)

Short Answer

Expert verified
0.18

Step by step solution

01

Understand the Independence of Events

If two events, say E and F, are independent, the occurrence of one does not affect the occurrence of the other. Mathematically, this means that the probability of both events happening simultaneously, denoted as \(P(E \text{ and } F)\), is the product of the probabilities of each event occurring individually.
02

Write Down the Given Probabilities

The problem states that \(P(E) = 0.3\) and \(P(F) = 0.6\). These are the probabilities of events E and F occurring independently.
03

Apply the Multiplication Rule for Independent Events

Use the multiplication rule for independent events: \(P(E \text{ and } F) = P(E) \times P(F)\). Substitute the given probabilities into this formula.
04

Perform the Multiplication

Calculate \(P(E \text{ and } F) = 0.3 \times 0.6\). This results in \(P(E \text{ and } F) = 0.18\).

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Key Concepts

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

probability multiplication rule
To easily understand how to find the probability of two events happening together, we use the probability multiplication rule. This rule states that if we are dealing with independent events, the probability of both events occurring at the same time is found by multiplying their individual probabilities.
  • For instance, if we have event E and event F, and they do not affect each other, we say they are independent.
  • This leads us to use the formula:
    \( P(E \text{ and } F) = P(E) \times P(F) \)
Think of it as combining two separate chances into one overall chance. It simplifies the process and provides a clear method to follow.
independent events definition
Understanding the definition of independent events is key for solving probability problems involving the probability multiplication rule.
Independent events are events where the outcome or occurrence of one event does not affect the outcome or occurrence of the other.
  • For example, flipping a coin and rolling a die are independent events. The result of the coin flip does not influence the result of the die roll.
  • Mathematically, if E and F are independent events, then: \(P(E|F) = P(E) \) and \(P(F|E) = P(F) \)
Recognizing independent events in problems helps us apply the multiplication rule correctly and efficiently.
probability calculation steps
Here is a detailed explanation of the steps to calculate the probability using the provided example:
  • Step 1: Recognize Independence - Understand that events E and F are independent, meaning their occurrence does not affect each other.
  • Step 2: Note Given Probabilities - Write down the given probabilities: \(P(E) = 0.3\) and \(P(F) = 0.6\).
  • Step 3: Apply Rule - Use the multiplication rule for independent events: \(P(E \text{ and } F) = P(E) \times P(F) \)
  • Step 4: Calculate - Substitute the given probabilities into the formula:
    \(P(E \text{ and } F) = 0.3 \times 0.6 = 0.18\)
Breaking down the steps and sequentially applying the known rules simplifies the process and ensures accuracy in solving probability problems.

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Most popular questions from this chapter

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