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Chapter 5: Problem 9

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

Short Answer

Expert verified
The probability of both events E and F occurring is 0.18.

Step by step solution

01

- Understand the Concept of Independence

For two events to be independent, the occurrence of one event does not affect the occurrence of the other. Mathematically, if events E and F are independent, then the probability of both events happening, denoted as \(P(E \cap F) \) or \(P(E \text{and} F)\), is given by \ P(E \cap F) = P(E) \cdot P(F) \.
02

- Write Down Given Probabilities

The problem states that \(P(E) = 0.3\) and \(P(F) = 0.6\). We will use these values in the next step.
03

- Calculate the Joint Probability

Since events E and F are independent, we use the formula: \(P(E \cap F) = P(E) \cdot P(F)\). Substituting the given probabilities: \(P(E \cap F) = 0.3 \cdot 0.6\).
04

- Perform the Multiplication

Calculate the product: \(0.3 \cdot 0.6 = 0.18\). Therefore, \(P(E \cap F) = 0.18\).

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Key Concepts

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

Joint Probability
Joint probability refers to the probability of two events happening at the same time. This is denoted as \( P(E \cap F) \) or \( P(E \text{ and } F) \). To find this probability, you need to understand the nature of events in question.
A good way of thinking about this is asking: 'What are the chances that both bubbles will pop together if each has its own popping chance?'.
It's important to note that Joint Probability is different from simple probability which looks at a single event's occurrence.
The joint probability combines both, giving you a fuller picture.
Independence of Events
Events are considered independent if the occurrence of one event does not influence the occurrence of another event. In simple terms, if flipping a coin doesn鈥檛 affect rolling a die, then these two events are independent. Mathematically, this is expressed as:
If \( E \) and \( F \) are independent, then \( P(E \cap F) = P(E) \cdot P(F) \).
This means the chances of both events happening together is just the product of their individual probabilities.
In our example, with \( P(E) = 0.3 \) and \( P(F) = 0.6 \), the probability that both events occur is calculated by multiplying these probabilities.
Multiplication Rule
The multiplication rule is essentially the tool we use for finding the joint probability of two independent events. When it鈥檚 given that events \( E \) and \( F \) are independent, the multiplication rule states that:
\( P(E \cap F) = P(E) \cdot P(F) \).
In our exercise, we substitute the known values \( P(E) = 0.3 \) and \( P(F) = 0.6 \) to find:
\( P(E \cap F) = 0.3 \cdot 0.6 = 0.18 \).
This final number, 0.18, represents the joint probability, the likelihood of both events happening together. It's a fundamental concept in probability and statistics, used to determine how likely it is for multiple events to occur simultaneously.

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Most popular questions from this chapter

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