Question

In a probability model, which of the following numbers could be the probability of an outcome? $$\begin{array}{llllll}0 & 0.01 & 0.35 & -0.4 & 1 & 1.4\end{array}$$

Short answer

The valid probabilities are 0, 0.01, 0.35, and 1.

Step-by-step solution

Understanding the Probability Range

In probability theory, the probability of any event ranges from 0 to 1, inclusive. This means that any valid probability value must lie within this range.

Checking Each Given Number

Examine each given number to determine if it lies within the range [0, 1]. The given numbers are 0, 0.01, 0.35, -0.4, 1, 1.4:- 0: Lies within the range.- 0.01: Lies within the range.- 0.35: Lies within the range.- -0.4: Does not lie within the range (it is negative).- 1: Lies within the range.- 1.4: Does not lie within the range (it is greater than 1).

Compiling the Valid Probabilities

List all the numbers that lie within the range [0, 1]. These numbers are 0, 0.01, 0.35, and 1.

Key concepts

Probability Range

In probability theory, one of the foundational ideas is understanding the range of valid probabilities.
Probability quantifies the likelihood of an event occurring and is always represented by numbers.
Importantly, these numbers fall within a specific range, known as the probability range.
This range is from 0 to 1, inclusive, meaning any valid probability must be between these two points, including 0 and 1 themselves.

Why is the range limited to 0 and 1?
- If an event has a probability of 0, it means it is impossible for this event to occur.
- If an event has a probability of 1, it means it is certain that this event will occur.
For any probability values between 0 and 1, the likelihood of the event occurring falls somewhere between impossible and certain.

Valid Probability Values

Once we understand the range in which probabilities must fall, we can evaluate whether specific values are valid.
In the given exercise, we are provided with several numbers, and we need to check if they fall within the range [0, 1].

Here's how to determine if a number is a valid probability value:

• 0: This lies within the range, making it a valid probability.
• 0.01: This lies within the range too, so it is valid.
• 0.35: Also lies within the range and is valid.
• -0.4: This number is negative, which falls outside the range, making it invalid.
• 1: This is within the range and is therefore valid.
• 1.4: This number is greater than 1, making it invalid.

From this, we see that only the values 0, 0.01, 0.35, and 1 are valid probabilities.

Inclusive Range in Probability

Understanding the concept of an inclusive range is important in probability theory.
When we say the probability range is inclusive, we mean that the endpoints of the range, 0 and 1, are included as valid values.
This distinction is vital because sometimes in mathematics or statistics, ranges are exclusive, meaning the endpoints would not be considered.

For probability, the inclusive range indicates:
- 0 is the minimum possible value for a probability and it indicates an impossible event.
- 1 is the maximum possible value for a probability and it indicates a certain event.

Any value within 0 and 1, including 0 and 1 themselves, is part of the valid set of probabilities.
This inclusivity ensures we can accurately represent events that are impossible or certain, alongside events with partial likelihoods.