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Chapter 5: Problem 4

Suppose that \(E\) and \(F\) are two events and that \(P(E \text { and } F)=0.21\) and \(P(E)=0.4 .\) What is \(P(F | E) ?\)

Short Answer

Expert verified
0.525

Step by step solution

01

- Understand the given information

Identify the given probabilities. Here, we have: \( P(E \text{ and } F) = 0.21 \) and \( P(E) = 0.4 \).
02

- Recall the conditional probability formula

The formula for conditional probability is given by: \[ P(F | E) = \frac{P(E \text{ and } F)}{P(E)} \]
03

- Substitute the given values into the formula

Substitute \( P(E \text{ and } F) = 0.21 \) and \( P(E) = 0.4 \) into the formula from Step 2: \[ P(F | E) = \frac{0.21}{0.4} \]
04

- Perform the division

Now, perform the division to find \( P(F | E) \): \[ P(F | E) = 0.525 \]

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Key Concepts

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

Probability
Probability is a measure of the likelihood of an event happening. It ranges from 0 to 1, where 0 means the event will not happen, and 1 means it definitely will. In our problem, the probability of event E, represented as P(E), is given as 0.4. This means there is a 40% chance that event E will occur. Understanding probabilities and how they work is fundamental to solving problems involving chance or uncertainty.
Events
In probability theory, we deal with events. An event is a set of outcomes to which we assign a probability. In this problem, we are given two events, E and F. Events can occur independently or dependently, affecting the calculations we perform. Here, we know how likely it is to have both events E and F happening together, which is given by P(E and F) = 0.21. Comprehending the relationships between events is crucial for correctly applying probability formulas.
Formula Substitution
To solve for conditional probability, we use a specific formula. Conditional probability measures the probability of an event occurring, given that another event has already happened. The formula we use is: \[ P(F | E) = \frac{P(E \text{ and } F)}{P(E)} \] Here, \( P(F | E) \) denotes the probability of event F occurring given that event E has occurred. By substituting the known probabilities into this formula, we can find the desired conditional probability.
Division
After substituting the known values into the conditional probability formula, we are left with a fraction that requires simplification: \[ P(F | E) = \frac{0.21}{0.4} \] Division is the operation we use to simplify this fraction. Breaking it down, it means we divide 0.21 by 0.4. Performing the division: \[ \frac{0.21}{0.4} = 0.525 \] Hence, the conditional probability \( P(F | E) \) equals 0.525, indicating there is a 52.5% chance of event F occurring, given that event E has already taken place.

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Most popular questions from this chapter

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