Answered

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



If an event cannot occur, its probability is -1.



Choose the correct answer below.

AThe statement is false. The true statement is "If an event cannot occur, its probability is 1."
BThe statement is false. The true statement is "If an event cannot occur, its probability cannot be found."
CThe statement is false. The true statement is "If an event cannot occur, its probability is 0."
DThe statement is true because the probability is -1 if an event cannot occur.

Verified Answer :

  • The range of any probability is \(0 \leq P(E) \leq 1\).
  • An impossible event has a probability of exactly \(0\).
  • The statement "If an event cannot occur, its probability is -1" is false.
  • The correct statement is: "The statement is false. The true statement is 'If an event cannot occur, its probability is 0'."
  • Final Answer: \(\boxed{C}\)

Explanation

  1. Understand the fundamental axioms of probability

The fundamental principle of probability theory states that the probability of any event \(E\), denoted as \(P(E)\), must lie within the range of \(0\) to \(1\), inclusive. That is, \(0 \leq P(E) \leq 1\). This range is a core axiom of probability.

  1. Define an impossible event and its probability

An impossible event is an event that contains no outcomes from the sample space. Because an impossible event cannot occur, it is assigned a probability of exactly \(0\). Therefore, if an event cannot happen, its probability is not negative; it is zero.

  1. Evaluate the statement

The given statement claims that if an event cannot occur, its probability is \(-1\). Comparing this to our axiom (\(0 \leq P(E) \leq 1\)), we can see that \(-1\) is outside the allowed range for any probability. Furthermore, our definition states that an impossible event has a probability of \(0\). Thus, the statement is false.

  1. Select the correct option

Looking at the given options:

  • A: Incorrect, as a probability of \(1\) corresponds to a certain event.
  • B: Incorrect, as the probability of an impossible event is defined as \(0\).
  • C: Correct, as it identifies the statement as false and provides the accurate definition ("the probability is \(0\)").
  • D: Incorrect, as a probability cannot be negative.

Therefore, option C is the correct choice.

Examples

In everyday life, we encounter "impossible" situations, such as rolling a 7 on a standard six-sided die. In mathematics, we quantify the likelihood of this event using probability. By knowing that the probability of this impossible event is \(0\), we can accurately model real-world scenarios and ensure that our statistical calculations remain within the valid range of \(0\) to \(1\).

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