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Determine whether the statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.



The sum of the probabilities of all possible outcomes in an experiment is 1.



Choose the correct answer below.

AThis statement is false. To make this statement true, change "1" to "4".
BThis statement is true.
CThis statement is false. To make this statement true, change "1" to "3".
DThis statement is false. To make this statement true, change "1" to "2".

Verified Answer :

  • The sum of all probabilities in a sample space equals 1 according to the Kolmogorov axioms.
  • The statement provided is a fundamental truth in probability theory.
  • The correct choice is B.
  • Final Answer: \(\boxed{B}\)

Explanation

  1. Analyze the fundamental principle of probability theory

The problem asks to evaluate the truthfulness of the statement: "The sum of the probabilities of all possible outcomes in an experiment is 1." This is a foundational concept in probability theory. In any random experiment, the set of all possible outcomes is called the sample space, denoted by \(S\). According to the Kolmogorov axioms, which provide the mathematical foundation for probability theory, the probability of the entire sample space \(P(S)\) must be equal to 1. Since the sample space is composed of all possible disjoint elementary outcomes, the sum of their individual probabilities must equal the probability of the entire space. Therefore, the sum of the probabilities of all possible outcomes in an experiment is indeed 1.

  1. Evaluate the provided options and conclude

Having confirmed that the sum of the probabilities of all possible outcomes in an experiment is defined as 1 by the axioms of probability, we evaluate the given options:

  • Option A: Claims the statement is false and suggests changing 1 to 4.
  • Option B: Claims the statement is true.
  • Option C: Claims the statement is false and suggests changing 1 to 3.
  • Option D: Claims the statement is false and suggests changing 1 to 2.

Since the statement is fundamentally true, Option B is the correct choice.

Examples

If you have a fair, six-sided die, the possible outcomes are {1, 2, 3, 4, 5, 6}. Each outcome has a probability of \(\frac{1}{6}\). The sum of these probabilities is \(\frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} = \frac{6}{6} = 1\). This example illustrates that for any discrete experiment, the total probability across all outcomes must always sum to 1.

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