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Chapter 5: Problem 3

Suppose that \(E\) and \(F\) are two events and that \(P(E \text { and } F)=0.6\) and \(P(E)=0.8 .\) What is \(P(F | E) ?\)

Short Answer

Expert verified
The conditional probability \(P(F | E)\) is 0.75.

Step by step solution

01

Understand the Problem

We are given the probability of two events E and F happening together, which is denoted as \(P(E \text{ and } F)=0.6\), and the probability of event E happening, \(P(E)=0.8\). We need to find the conditional probability of F given E, which is denoted as \(P(F | E)\).
02

Recall the Formula for Conditional Probability

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

Substitute the Given Values

Using the values provided in the problem, substitute \(P(E \text{ and } F)\) with 0.6 and \(P(E)\) with 0.8 into the formula:\[ P(F | E) = \frac{0.6}{0.8} \]
04

Simplify the Fraction

Simplify the fraction \(\frac{0.6}{0.8}\): \[ P(F | E) = 0.75 \]

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Key Concepts

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

Probability Theory
Probability theory is the branch of mathematics that deals with the likelihood of different events happening. It uses numbers between 0 and 1 to express this likelihood, where 0 means the event will not happen, and 1 means it definitely will. Probability theory is fundamental in understanding and predicting the behavior of systems affected by randomness.
An example could be the probability of rolling a 3 on a six-sided die, which is \(\frac{1}{6}\) because one out of six possible outcomes is a 3. Probability plays a crucial role in various fields like finance, science, engineering, and even everyday decision-making.
Events
An event in probability theory refers to a specific outcome or a set of outcomes of a random experiment. For example, if you roll a die, getting an even number (2, 4, or 6) is considered an event.

Events can be either:
  • Simple events: Single outcomes like rolling a 3 on a die.
  • Compound events: Combinations of two or more simple events, like rolling an even number.
Understanding events helps in calculating probabilities, especially when dealing with combinations and permutations, such as in the exercise where two events E and F are given.
Conditional Probability Formula
Conditional probability is the probability of an event occurring given that another event has already occurred. This is denoted by \(P(F | E)\) and is read as 'the probability of F given E'.

The formula for conditional probability is:
\[ P(F | E) = \frac{P(E \text{ and } F)}{P(E)} \]
This means you divide the probability of both events E and F happening together by the probability of event E. In simpler terms, you adjust the probability of F by taking into account that E has already happened.
For example, in the exercise provided, if you know event E has occurred with a probability of 0.8 and events E and F together have a probability of 0.6, you use the formula to find \(P(F | E)\).
Simplifying Fractions
Understanding how to simplify fractions is crucial in probability, especially when dealing with conditional probabilities. Simplifying a fraction means reducing it to its simplest form.
In the exercise, you need to simplify \(\frac{0.6}{0.8}\).

Steps to simplify:
  • Find the greatest common divisor (GCD): Here, GCD of 0.6 and 0.8 is 0.2.
  • Divide both numerator and denominator by the GCD: \(\frac{0.6 \div 0.2}{0.8 \div 0.2} = \frac{3}{4}\).
  • Result: The simplified fraction is 0.75.
This process helps in getting clean, easy-to-understand results when working with probabilities.

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Most popular questions from this chapter

In the Illinois Lottery game Little Lotto, an urn contains balls numbered 1 to \(30 .\) From this urn, 5 balls are chosen randomly, without replacement. For a \(\$ 1\) bet, a player chooses one set of five numbers. To win, all five numbers must match those chosen from the urn. The order in which the balls are selected does not matter. What is the probability of winning Little Lotto with one ticket?

Suppose a single card is selected from a standard 52 -card deck. What is the probability that the card drawn is a club? Now suppose a single card is drawn from a standard 52 -card deck, but we are told that the card is black. What is the probability that the card drawn is a club?

Exclude leap years from the following calculations and assume each birthday is equally likely: (a) Determine the probability that a randomly selected person has a birthday on the 1 st day of a month. Interpret this probability. (b) Determine the probability that a randomly selected person has a birthday on the 31 st day of a month. Interpret this probability. (c) Determine the probability that a randomly selected person was born in December. Interpret this probability. (d) Determine the probability that a randomly selected person has a birthday on November \(8 .\) Interpret this probability. (e) If you just met somebody and she asked you to guess her birthday, are you likely to be correct? (f) Do you think it is appropriate to use the methods of classical probability to compute the probability that a person is born in December?

Suppose 40 cars start at the Indianapolis \(500 .\) In I how many ways can the top 3 cars finish the race?

List all the combinations of five objects \(a, b, c, d,\) and \(e\) taken two at a time. What is \(_5 C_{2} ?\)
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