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Chapter 14: Problem 136

If \(P(E)=.3\) and \(P(F)=.4\) and \(E\) and \(F\) are independent, find \(P(E \mid F)\)

Short Answer

Expert verified
0.3

Step by step solution

01

Understanding Independence

Two events E and F are said to be independent if the occurrence of one does not change the probability of the occurrence of the other. In mathematical terms, this means P(E | F) = P(E).
02

Identify Relevant Probabilities

From the exercise, we have two relevant probabilities which are P(E) = 0.3 and P(F) = 0.4.
03

Calculating Conditional Probability

For independent events, the conditional probability P(E | F) is simply P(E). Since we know P(E) = 0.3, we can state that P(E | F) = P(E) = 0.3.

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Key Concepts

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

Independent Events
In probability, the concept of independent events is quite fundamental. When we say two events are independent, we are talking about their unique characteristic where one event occurring does not affect the probability of the other event occurring. This is a crucial concept because it simplifies calculations.
For example, consider the flip of a fair coin or the roll of a fair die. Whether the coin lands on heads or tails does not influence the outcome of rolling a die. This independence allows us to compute individual probabilities without worrying about complicated interdependencies between events.
  • Independent events have no impact on each other.
  • Knowing the outcome of one tells us nothing new about the likelihood of the other.
  • Mathematically, if events E and F are independent, then \( P(E \mid F) = P(E) \).
This property plays a key role in many areas of probability and statistics.
Probability
Probability is a branch of mathematics that deals with the likelihood of events occurring. It is the basis for predicting future outcomes based on known factors and is expressed on a scale from 0 to 1. A probability of 0 means an event will not happen, and a probability of 1 means it will certainly happen.
  • The probability of an event occurring and not occurring always sums up to 1.
  • Probabilities are usually written as fractions, percentages, or decimals.
  • In our exercise, we are given two probabilities: \( P(E) = 0.3 \) and \( P(F) = 0.4 \).
Understanding probability helps in making calculated decisions under uncertainty and analyzing scenarios where uncertainty is involved.
Mathematics Education
Learning mathematics, especially probability, equips students with valuable skills for problem-solving and analytical thinking. Mathematics education emphasizes understanding concepts deeply, rather than just memorizing formulas and procedures.
For instance, the exercise on independent events and conditional probabilities not only teaches calculation but also illustrates how to think about relationships between different events.
  • Students gain confidence by solving practical problems where the outcome is not entirely certain.
  • Mathematics education encourages logical reasoning and critical thinking.
  • Problem-based learning, like solving conditional probability exercises, helps in applying theoretical knowledge to real-world situations.
Through continuous practice, students develop a profound understanding of concepts like independence, enhancing their ability to tackle complex mathematical challenges.

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Most popular questions from this chapter

Do the following problems using the conditional probability formula: \(P(A \mid B)=\frac{P(A \cap B)}{P(B)}\). If \(P(A)=.3\) and \(P(B)=.4,\) and \(P(A\) and \(B)=.12,\) find the following. a. \(P(A \mid B)\) b. \(P(B \mid A)\)

Consider a family of three children. Find the following probabilities. P (two boys and a girl)

Consider a family of three children. Find the following: a. \(P(\) children of both sexes \(\mid\) first born is a boy) b. \(P\) (all girls | children of both sexes)

Again, use the addition rule to determine the following probabilities. If \(P(E)=.4, P(F)=.5\) and \(P(E\) or \(F)=.7,\) find \(P(E\) and \(F)\)

Mrs. Rossetti is flying from San Francisco to New York. On her way to the San Francisco Airport she encounters heavy traffic and determines that there is a \(20 \%\) chance that she will be late to the airport and will miss her flight. Even if she makes her flight, there is a \(10 \%\) chance that she will miss her connecting flight at Chicago. What is the probability that she will make it to New York as scheduled?
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