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Magazine Chapter 5: Problem 12
Which of the following numbers could be the probability of an event? $$ 1.5, \frac{1}{2}, \frac{3}{4}, \frac{2}{3}, 0,-\frac{1}{4} $$
Short Answer
Expert verified
Valid probabilities: \(\frac{1}{2}\), \(\frac{3}{4}\), \(\frac{2}{3}\), 0.
Step by step solution
01
Understand the Concept of Probability
Probability values represent the likelihood of an event occurring and must lie between 0 and 1, inclusive. This means any valid probability must be greater than or equal to 0 and less than or equal to 1.
02
Evaluate Each Number
Check each given number to see if it falls within the range [0, 1].
03
Examine 1.5
1.5 is greater than 1, so it cannot be a probability.
04
Examine \(\frac{1}{2}\)
\(\frac{1}{2}\) is between 0 and 1, so it can be a probability.
05
Examine \(\frac{3}{4}\)
\(\frac{3}{4}\) is between 0 and 1, so it can be a probability.
06
Examine \(\frac{2}{3}\)
\(\frac{2}{3}\) is between 0 and 1, so it can be a probability.
07
Examine 0
0 is at the low end of the range, so it can be a probability.
08
Examine \(\-\frac{1}{4}\)
\(-\frac{1}{4}\) is less than 0, so it cannot be a probability.
09
List Valid Probability Values
The numbers that fall within the range [0, 1] and can be probabilities are \(\frac{1}{2}\), \(\frac{3}{4}\), \(\frac{2}{3}\), and 0.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Valid Probability Range
Probabilities measure how likely an event is to happen. They are always numbers within a specific range. This is because an event cannot happen less than never (0) and cannot happen more than always (1).
Thus, the valid range for probabilities is from 0 to 1, inclusive. This means:
They simply do not make sense in the context of likelihood.
Thus, the valid range for probabilities is from 0 to 1, inclusive. This means:
- 0 is the smallest probability, meaning the event will never happen
- 1 is the largest probability, meaning the event will surely happen
- Any number between these values (like 0.25 or 0.75) is also considered a probability
They simply do not make sense in the context of likelihood.
Evaluating Probabilities
When given a set of numbers, you can evaluate them to see if they can be valid probabilities.
Here's how you can do it step by step:
Here's how you can do it step by step:
- First, check if the number is greater than or equal to 0. If it's not, like -0.4, it's immediately invalid.
- Next, check if the number is less than or equal to 1. Numbers greater than 1, like 1.5, do not fit within the valid range.
- Finally, if the number fits both criteria, then it is a valid probability. For example, 0.5 and 0.75 fit the criteria and are valid.
Inclusive Ranges in Probability
The word 'inclusive' is crucial when talking about ranges in probability.
Inclusive means that the end values of the range (0 and 1) are part of the valid set of numbers. So, both 0 and 1 are valid probabilities:
This inclusive range helps to cover all possible probabilities for any event.
Inclusive means that the end values of the range (0 and 1) are part of the valid set of numbers. So, both 0 and 1 are valid probabilities:
- 0 is the probability that something will not happen at all.
- 1 is the probability that something will definitely happen.
This inclusive range helps to cover all possible probabilities for any event.
Probability Basics
Understanding the basics of probability starts with knowing why we use it.
Probability helps us measure and predict the likelihood of events, from rolling a dice to weather forecasts. Here are few key points to remember:
Probability helps us measure and predict the likelihood of events, from rolling a dice to weather forecasts. Here are few key points to remember:
- Probability values always lie between 0 and 1, inclusive.
- A probability of 0 means the event will never happen, while a probability of 1 means the event will always happen.
- Probabilities can be expressed as fractions, decimals, or percentages. For example, 0.5 is the same as 50% or \(\frac{1}{2}\).
