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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

A screening test for a certain disease is prone to giving false positives or false negatives. If a patient being tested has the disease, the probability that the test indicates a (false) negative is \(.13 .\) If the patient does not have the disease, the probability that the test indicates a (false) positive is .10. Assume that \(3 \%\) of the patients being tested actually have the disease. Suppose that one patient is chosen at random and tested. Find the probability that a. this patient has the disease and tests positive b. this patient does not have the disease and tests positive c. this patient tests positive d. this patient has the disease given that he or she tests positive (Hint: A tree diagram may be helpful in part c.)

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?

The probability that a farmer is in debt is 80 . What is the probability that three randomly selected farmers are all in debt? Assume independence of events.

How is the addition rule of probability for two mutually exclusive events different from the rule for two mutually nonexclusive events?

There are 142 people participating in a local \(5 \mathrm{~K}\) road race. Sixty-five of these runners are female. Of the female runners, 19 are participating in their first \(5 \mathrm{~K}\) road race. Of the male runners, 28 are participating in their first \(5 \mathrm{~K}\) road race. Are the events female and participating in their first \(5 \mathrm{~K}\) road race independent? Are they mutually exclusive? Explain why or why not.
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