Problem 6 - Algebra and Trigonometry

Question

In a probability model, which of the following numbers could be the probability of an outcome? $$\begin{array}{llll}1.5 & \frac{1}{2} & \frac{3}{4} & \frac{2}{3} & 0 & -\frac{1}{4}\end{array}$$

Step-by-Step Solution

Answer

Valid probabilities are: \[\frac{1}{2}\], \[\frac{3}{4}\], \[\frac{2}{3}\], and 0.

Step 1 - Understand Probability Values

Probability values are always between 0 and 1, inclusive. This means that any number less than 0 or greater than 1 cannot represent the probability of an outcome.

Step 2 - Evaluate Each Number

Check each given number to see if it falls within the range [0, 1].

Step 3 - Analyze the List of Numbers

1.5: This number is greater than 1, so it is not a valid probability.\[\frac{1}{2}\]: This number is between 0 and 1, so it is a valid probability.\[\frac{3}{4}\]: This number is between 0 and 1, so it is a valid probability.\[\frac{2}{3}\]: This number is between 0 and 1, so it is a valid probability.0: This number is 0, so it is a valid probability.\[-\frac{1}{4}\]: This number is less than 0, so it is not a valid probability.

Step 4 - List Valid Probabilities

The numbers that are valid probabilities are: \[\frac{1}{2}\], \[\frac{3}{4}\], \[\frac{2}{3}\], and 0.

Key Concepts

valid probabilities

In probability theory, probabilities measure how likely an event is to occur. For a probability to be valid, it must follow certain rules.

First, probabilities are always non-negative. This means values like \(-\frac{1}{4}\) do not qualify as valid probabilities because they are less than 0.

Second, probabilities cannot exceed 1. Any number greater than 1, such as 1.5, also fails to represent a probability of an event.

Lastly, probabilities must be bounded within the range from 0 to 1, inclusive. Therefore, numbers like \(\frac{1}{2}\), \(\frac{3}{4}\), \(\frac{2}{3}\), and even 0 are acceptable probabilities as they all fall within this range.

To summarize, for a probability to be valid, it must be [0, 1]. This means:

• It cannot be negative
• It cannot be greater than 1
• It can be any number between 0 and 1, including 0 and 1 themselves.

probability range

The probability range is critical in determining the validity of probability values. This range is expressed as [0, 1].

Every valid probability must lie within or on the boundaries of this interval. Let's explore this concept further.

• A probability of 0 indicates an impossible event. For instance, if you roll a fair die, the probability of rolling a seven is zero.
  
  
• A probability of 1 indicates a certain event. For example, the probability that the sun will rise tomorrow is essentially 1.
  
  
• Probabilities between 0 and 1 describe events that are neither impossible nor certain. A classic example is flipping a fair coin, where the probability of getting heads is \(\frac{1}{2}\).

Using the given numbers, numbers like 1.5 (greater than 1) and \(-\frac{1}{4}\) (less than 0) stand outside this range, making them invalid.

To determine valid probabilities, always check if the numbers fall within the [0, 1] range.

probability model

A probability model is a mathematical framework that defines all the possible outcomes of a random experiment and assigns a probability to each outcome.

Here are the key components of a probability model:

• Sample Space (S): This includes all the possible outcomes of the experiment. For instance, flipping a coin has a sample space of {Heads, Tails}.


• Event (E): An event is a subset of the sample space. For example, flipping a coin and landing on Heads is one event.


• Probability Function (P): This function assigns a probability to each event in the sample space. The function must satisfy two conditions:
  
  • For any event E, \(0 \leq P(E) \leq 1\)
  • The total probability of all outcomes in the sample space must equal 1, represented as:
  \[P(S) = 1\]

In our exercise, we evaluated numbers to determine if they could be valid probabilities. Only numbers within the [0, 1] range were accepted as valid. Therefore, valid probabilities ensure that the probability function remains consistent within the defined probability model.