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Chapter 4: Problem 15

Briefly explain the two properties of probability.

Short Answer

Expert verified
The two properties of probability are non-negativity, which states that a probability can never be a negative number, and finite additivity, which specifies that the sum of probabilities of all possible outcomes is always equal to one.

Step by step solution

01

Property 1: Non-Negativity

The first property of probability is Non-negativity. In layman's terms, this principle states that probability cannot be a negative number. In a mathematical formulation, if 'E' is an event and 'P(E)' is the probability of the event, then 0 鈮� P(E). This means that the smallest value that probability can take is zero. For instance, the chance of anyone living on Mars is zero because there is no life on Mars as far as we know.
02

Property 2: Finite Additivity

The second property of probability is Finite Additivity (also known as the Unit Measure property). It denotes that the sum of the probabilities of all possible outcomes always equals one. In terms of a formula, if 'S' is a sample space, then P(S) = 1. For example, If we flip a coin, the possibilities are heads or tails. The probability of getting a head is 0.5 and tail is also 0.5. Adding both probabilities would give a total of 1.

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Key Concepts

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

Non-Negativity
Probability is a way to measure how likely an event is to happen. One important rule is called **Non-Negativity**. Non-negativity means you can't have a negative chance of something happening, which makes sense! Let's break it down a little more.

Think of an event, like flipping a coin or drawing a card. The probability of any event happening is always a number between 0 and 1. It can't be less than 0. If you were told that the chance of drawing the Queen of Hearts is -0.2, you'd know something's off. That's because probability can never be negative!

  • If the probability of an event is 0, it means the event can't happen, just like the chance of it raining jelly beans is 0.
  • If the probability is 1, the event is certain to happen, such as the sun rising in the morning.
Finite Additivity
A second key principle of probability is called **Finite Additivity**. It tells us that when you add up the probabilities of all possible outcomes of a particular event, the total should equal 1. This helps to ensure that every possible outcome is accounted for.

Imagine you have a simple six-sided die. Each side has numbers from 1 to 6. When you roll the die, only one number will land face up. The total probability of landing on any one of these numbers (1 to 6) should be 1. Here's why:
  • The probability of rolling a 1 is \( \frac{1}{6} \).
  • Similarly, the probability of rolling any of the numbers (2 through 6) is also \( \frac{1}{6} \).
  • If you add them all together, you get \( \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} = 1 \).
This means all possibilities are covered, making sure nothing is left out.
Sample Space
To understand probability, it's crucial to know what a **Sample Space** is. The sample space is the set of all possible outcomes in a probability experiment. Basically, it lists every possible outcome that could occur.
Consider a simple experiment: tossing a coin. The sample space for this experiment is {Heads, Tails}. Every time you flip the coin, you'll get one of these outcomes.

  • For a standard six-sided die, the sample space is {1, 2, 3, 4, 5, 6}.
  • In a survey where people can choose their favorite color from red, blue, green, the sample space is {Red, Blue, Green}.
Sample spaces are crucial in calculating probabilities because they help us understand what all the possible outcomes are, setting the stage for calculating how likely each one is.

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Most popular questions from this chapter

Find the joint probability of \(A\) and \(B\) for the following. a. \(P(A)=.36\) and \(P(B \mid A)=.87\) b. \(P(B)=.53\) and \(P(A \mid B)=.22\)

According to the Recording Industry Association of America, only \(37 \%\) of music files downloaded from Web sites in 2009 were paid for. Suppose that this percentage holds true for such files downloaded this year. Three downloaded music files are selected at random. What is the probability that all three were paid for? What is the probability that none were paid for? Assume independence of events.

An investor will randomly select 6 stocks from 20 for an investment. How many total combinations are possible? If the order in which stocks are selected is important, how many permutations will there be?

A company employs a total of 16 workers. The management has asked these employees to select 2 workers who will negotiate a new contract with management. The employees have decided to select the 2 workers randomly. How many total selections are possible? Considering that the order of selection is important, find the number of permutations.

Find \(P(A\) or \(B\) ) for the following. a. \(P(A)=.66, \quad P(B)=.47\), and \(P(A\) and \(B)=.33\) b. \(P(A)=.84, \quad P(B)=.61\), and \(P(A\) and \(B)=.55\)
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