SciTech-Mathematics-Probability+Statistics-Comparison:Chance + Possibility + Likelihood + Probability

https://www.geeksforgeeks.org/what-is-the-difference-between-likelihood-and-probability/

1. Chance

2. Possibility

3. Likelihood

4. Probability

In summary:

• \(\large probability\) quantifies the \(\large likelihood\) of future events.
• while \(\large likelihood\) quantifies the \(\large probability\) of past observations given a specific model or hypothesis.

Understanding the distinction between these concepts is crucial for conducting \(\large statistical\ inference\) and interpreting the results of \(\large statistical\ analyses\) accurately.

Difference: "Likelihood" and "Probability"

Last Updated : 30 Jul, 2024

https://www.geeksforgeeks.org/what-is-the-difference-between-likelihood-and-probability/

Answer: In statistics,

• comparing to "Resolve" and "Determination":
  
  Likelihood similar to Resolve, Probability similar to Determination
• "likelihood": refers to the chance of observing data given a particular model or hypothesis.
• "probability": represents the chance of an event(of a Experiment) occurring a head of time.

Likelihood vs Probability: Comparison

Feature	Likelihood	Probability
Definition	The probability of observing data given a specific model or hypothesis.	The measure of the likelihood that an event will occur before it happens.
Application	Used in statistical inference to assess the plausibility of different parameter values given observed data.	Used in probability theory to quantify uncertainty associated with the occurrence of future events.
Directionality	Backward-looking: concerns the probability of past observations given a model.	Forward-looking: concerns the likelihood of future events.
Parameterization	Associated with the likelihood of parameter values given observed data.	Associated with the likelihood of outcomes of random experiments or events.
Interpretation	Interpreted as a measure of support for different parameter values given observed data.	Interpreted as a measure of belief or uncertainty about future events.
Example	In \(\large linear\ regression\), the \(\large likelihood\ function\) measures the \(\large probability\) of observing the given set of data points under the assumption that they are generated from a linear relationship between the variables.	The \(\large probability\) of \(\large \text{rolling a six on a fair six-sided}\ die\) is 1661​ because there is one favorable outcome (rolling a six) out of six equally likely possible outcomes.

Similar Questions

Q1. What is the difference between likelihood and probability in statistical analysis?

Q2. How does likelihood relate to parameter estimation in statistical models?

Q3. Can you provide another example of how likelihood is used in statistical inference?

Q4. How is probability used to make predictions about future events?

Q5. What role does probability play in hypothesis testing?

Q6. How does Bayesian inference use the concept of likelihood?

Q7. What is the likelihood function in the context of maximum likelihood estimation?

Q8. How do likelihood and probability differ in their application to random experiments?

Q9. Can you explain the role of likelihood in model selection and comparison?

Q10. How does the concept of probability apply to everyday decision-making scenarios?

Difference: Likelihood vs. Probability

BY ZACH BOBBITTPOSTED ON AUGUST 18, 2021

Two terms that students often confuse in statistics are likelihood and probability.

THREE STEPS TO SOLVE A PROBLEM WITH STATISTICAL MODEL:

1. Modeling:

   • Initialization: Select A Model from a list of available models.
   • Conditions
   • Parameterization: Parameters, Types, Ranges, Values, Limit of Permissible Variation.

2. Verification of Model:

   • Observation: Make a Sample from observations of the selected model and Parameters.
   • Verification: trying to determine if we can trust the Parameters and the Model, from the Sample Data we have observed.
   • we can identify the possible outcomes for next step: Experiment and Event.
     
     Observation and Verification makes possible outcomes are sure.
   • it would be desirable to be able to make A Precise Statement of the Likelihoods of the Different Possible numbers of the each outcomes.

3. Probability+Statistics

   • Experiment and Event
   • Assign Probabilities to possible outcomes.

Here's the difference in a nutshell:

• Probability refers to the chance that a particular outcome occurs based on the values of parameters in a model.
  
  When calculating the probability of some outcome, we assume the parameters in a model are trustworthy.

• Likelihood refers to how well a Sample Data provides Support for particular values of a parameter in a model.
  
  However, when we calculate likelihood we're trying to determine if we can trust the parameters in a model based on the sample data that we've observed.

The following examples illustrate the difference between probability and likelihood in various scenarios.

Example 1: Likelihood vs. Probability in Coin Tosses

Suppose(Modeling):

• we have a coin that is assumed to be fair.

• If we flip the coin one time, the probability that it will land on heads is 0.5.
  
  Now suppose(Model Verification):
  
  we flip the coin 100 times and it only lands on heads 17 times.
• We would say that the likelihood that the coin is fair is quite low.
  
  If the coin was actually fair, we would expect it to land on heads much more often.
• When calculating the probability of a coin landing on heads,
  
  we simply assume that P(heads) = 0.5 on a given toss.
• When calculating the likelihood we're trying to determine if the model parameter (p = 0.5) is actually correctly specified.

In the example above, a coin landing on heads only 17 out of 100 times,

makes us highly suspicious that,

the truly probability of the coin landing on heads on a given toss is actually p = 0.5.

Example 2: Likelihood vs. Probability in Spinners

Suppose we have a spinner split into thirds with three colors on it: red, green, and blue.

Suppose we assume that it's equally likely for the spinner to land on any of the three colors.

If we spin it one time, the probability that it lands on red is 1/3.

a spinner split into thirds with three colors on it: red, green, and blue.

Now suppose we spin it 100 times and it lands on red 2 times, green 90 times, and blue 8 times. * * We would say that the likelihood that "the spinner is actually equally likely to land on each color" is very low.

• When calculating the probability of the spinner landing on red, we simply assume that P(red) = 1/3 on a given spin.
• When calculating the likelihood we're trying to determine if the model parameters (P(red) = 1/3, P(green) = 1/3, P(blue) = 1/3) are actually correctly specified.

In the example above, the results of the 100 spins make us highly suspicious that each color is equally likely to occur.

Example 3: Likelihood vs. Probability in Gambling

Suppose a casino claims that the probability of winning money on a certain slot machine is 40% for each turn.

If we take one turn, the probability that we will win money is 0.40.

Now suppose we take 100 turns and we win 42 times. We would conclude that the likelihood that the probability of winning in 40% of turns seems to be fair.

• When calculating the probability of winning on a given turn, we simply assume that P(winning) =0.40 on a given turn.
• When calculating the likelihood we're trying to determine if the model parameter P(winning) = 0.40 is actually correctly specified.

In the example above, winning 42 times out of 100 makes us believe that a probability of winning 40% of the time seems reasonable. ( Believe? )

Additional Resources

The following tutorials provide addition information about probability:

What is a Probability Distribution Table?

What is the Law of Total Probability?

How to Find the Mean of a Probability Distribu