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

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.**Likelihood**refers to how well a sample provides support for particular values of a parameter in a model.

When calculating the probability of some outcome, we assume the parameters in a model are trustworthy.

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 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.

**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 Distribution

How to Find the Standard Deviation of a Probability Distribution

Thanks!

Great!

Excellent expliation