It is essential to note that while the two terms may seem similar, they have a partial overlap but cannot be used interchangeably.
In fact, ‘likelihood’ refers to the degree of belief or expectation that something will happen, while ‘probability’ refers to the ratio of favorable outcomes to the total number of possible outcomes. Therefore, it is crucial to understand the subtle differences between these two concepts in order to use them correctly in different contexts. For instance, you would use probability when calculating the chances of winning a lottery. Still, you would use likelihood when assessing the likelihood of a specific event based on available evidence.
The likelihood is the chance, the possibility of doing or achieving something, and the condition that can ensure success. Probability is a degree, a relative measure, or a quantitative assessment of the possibility of an event occurring.
The probability is the ratio of desired outcomes to all possible outcomes, but the likelihood is the ratio of the possibility of an event to the probability of the absence of an event. 1
Probability corresponds to finding the chance of something given a sample distribution of the data, while on the other hand, Likelihood refers to finding the best distribution of the data given a particular value of some feature or some situation in the data. 2
Two terms that students often confuse in statistics are likelihood and probability. Here’s the difference in a nutshell: 3
- Probability refers to the chance that a particular outcome occurs based on the values of parameters in a model.
- Likelihood refers to how well a sample provides support for particular values of a parameter in a model.
Examples
The following examples are taken from the STATOLOGY webpage “Likelihood vs. Probability: What’s the Difference?”.
Example 1: Likelihood vs. Probability in Coin Tosses
Suppose 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 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.
However, 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.
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.
However, 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.
However, 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.
Probability Vs Likelihood
| Probability | Likelihood |
|---|---|
| It expresses the likelihood of an event taking place given a certain set of circumstances. | It describes the likelihood of a set of circumstances in light of an observed event. |
| It stands for a number between 0 and 1, where 0 denotes impossible and 1 denotes assurance. | It is a function of a number of factors rather than having a finite numerical range. |
| It can be applied to predict the likelihood of an occurrence occurring frequently in the future. | You can use it to draw conclusions about the circumstances that contributed to an observed occurrence. |
| Using the equation P(event) = (number of positive outcomes) / 2, it can be determined (total number of possible outcomes) | It can be determined by applying the equation L(conditions) = P(event). |
| The calculation needs a set of criteria that are known. | To compute, a witnessed event is necessary. |
| Prior information or presumptions about the situation or circumstances can have an impact. | It can be impacted by the selection of parameters or presumptions regarding the underlying process. |
| It can be employed to figure out what a random variable’s anticipated value is. | It cannot be used to calculate a random variable’s anticipated value. |
| Once fresh data is gathered, it is utilized in Bayesian statistics to update beliefs. | To identify the set of circumstances that maximizes the likelihood of the observed data, maximum likelihood estimation is performed. |
References
1 
Gusarova, Maria. “The differences between likelihood and probability — simply explained with examples”. 2023. Medium. https://medium.com/@data.science.enthusiast/the-differences-between-likelihood-and-probability-simply-explained-with-examples-7fed16aff61f.
2 Dawar, Harshit. “Probability VS Likelihood”. 2020. Medium. https://medium.com/swlh/probability-vs-likelihood-cdac534bf523.
This blog aims to explain the difference between the Probability & the Likelihood. This topic is very important to understand, but the problem here is that both the topics are very confusing to understand. That is why, I am writing this blog to remove the confusion, & I will explain the topics in a simple manner as possible.
3 Zach. “Likelihood vs. Probability: What’s the Difference?”. 2021. STATOLOGY. https://www.statology.org/likelihood-vs-probability/.
Additional Reading
“Difference Between Probability and Likelihood”. 2023. tutorialspoint.com. https://www.tutorialspoint.com/difference-between-probability-and-likelihood.
“What is the difference between Probability and Likelihood? | i2tutorials”. 2019. i2tutorials. https://www.i2tutorials.com/what-is-the-difference-between-probability-and-likelihood/.
Videos
Here’s one of those tricky little things, Probability vs. Likelihood. In common conversation we use these words interchangeably. However, statisticians make a clear distinction that is important to understand if you want to follow their logic. The good news is that they are both super simple. The bad news is that they are easy to get mixed up. The StatQuest gives you visual images that make them both easy to remember so you’ll always keep them straight.
NOTE: This video was originally made as a follow up to an overview of Maximum Likelihood. That video provides context that gives this video more meaning.
The featured image on this page is from the What is the difference between Probability and Likelihood? page on the i2tutorials website.
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