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Chapter 5: Problem 19

If \(P(E)=0.60, P(E \text { or } F)=0.85,\) and \(P(E \text { and } F)=0.05\) find \(P(F)\)

Short Answer

Expert verified
P(F) = 0.30

Step by step solution

01

Understand the Given Probabilities

Identify the probabilities given in the problem: \(P(E) = 0.60\), \(P(E \text{ or } F) = 0.85\), and \(P(E \text{ and } F) = 0.05\).
02

Use the Formula for Union of Two Events

Use the formula for the probability of the union of two events: \[ P(E \text{ or } F) = P(E) + P(F) - P(E \text{ and } F) \].
03

Substitute the Given Probabilities

Substitute the given probabilities into the formula: \[ 0.85 = 0.60 + P(F) - 0.05 \].
04

Simplify the Equation

Simplify the equation to solve for \(P(F)\): \[ 0.85 = 0.55 + P(F) \].
05

Solve for \(P(F)\)

Isolate \(P(F)\) by subtracting 0.55 from both sides of the equation: \[ P(F) = 0.85 - 0.55 \].
06

Calculate the Result

Perform the subtraction to find \(P(F)\): \[ P(F) = 0.30 \].

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Key Concepts

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

Union of Events
In probability theory, the union of two events represents the scenario where at least one of the events occurs. This can be written as \(P(E \text{ or } F)\). To calculate the union's probability, we use the formula: \(P(E \text{ or } F) = P(E) + P(F) - P(E \text{ and } F)\). This formula ensures that we don't double-count the overlap between the two events. In our exercise, substituting the given values in the union formula helped us find the unknown probability of event F.
Conditional Probability
Conditional probability measures the probability of an event occurring, given that another event has already occurred. It is denoted as \(P(A \mid B)\), which is read as 'the probability of A given B.' However, our exercise doesn't explicitly involve conditional probability, but it's a crucial concept to understand deeper probability calculations. It often uses the formula: \[P(A \mid B) = \frac{P(A \text{ and } B)}{P(B)}\]. This formula is key for many advanced probability questions.
Event Intersection
The intersection of two events occurs when both events happen simultaneously. It is denoted as \(P(E \text{ and } F)\). This concept is essential for calculating the overlap between events, ensuring that probabilities are correctly combined. In our exercise, \(P(E \text{ and } F) = 0.05\). This value played a critical role in our calculation of \(P(F)\) by being subtracted from the sum of \(P(E) \text{ and } P(F)\) to prevent double counting in the union formula.
Probability Formulae
Probability formulae are the backbone of solving probability problems. They provide a structured way to calculate different scenarios. The key formula used in the exercise is: \[P(E \text{ or } F) = P(E) + P(F) - P(E \text{ and } F)\]. By substituting the given values, we simplify it step-by-step to isolate and determine the unknown probability. Knowing these formulae and how they interconnect can simplify complex probability calculations.

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Most popular questions from this chapter

(a) Roll a single die 50 times, recording the result of each roll of the die. Use the results to approximate the probability of rolling a three. (b) Roll a single die 100 times, recording the result of each roll of the die. Use the results to approximate the probability of rolling a three. (c) Compare the results of (a) and (b) to the classical probability of rolling a three.

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