What’s the Difference: Likelihood vs Probability

Probability and likelihood are two terms students frequently confuse when studying statistics.

Likelihood vs Probability

This is the key difference:

• Probability is the probability that a certain outcome will occur based on the parameters of a model.
• The likelihood refers to the degree that which a sample supports particular values in a model.
• We assume that the parameters of a model are reliable when calculating the probability of a given outcome.
• We are trying to assess if the model’s parameters can be trusted based on the data we have observed. Try an online probability calculator for probability and find the probability calculator here.

These examples show the difference between likelihood and probability in different scenarios.

Example 1: Probability vs. Likelihood in Coin Tosses

Let’s say we have a coin that is fair. The probability of the coin landing on its head if we flip it once is 0.5.

Let’s say we flip the coin 100x and it only lands 17 times on heads. The probability that the coin is fair would be very low. If the coin were fair, it would land on heads more often.

We assume that P(heads = 0.5) on a given coin to calculate the probability of it landing on heads when calculating the probability of it landing on heads.

When calculating the likelihood, however, we are trying to determine if the model parameter p = 0.5 is correctly specified.

The fact that a coin lands on heads 17 times out of 100 makes it highly unlikely that a coin will land on heads in a given toss.

Example 2: Probability vs. Likelihood in Spinners

Let’s say we have a spinner that is divided into three parts with three colors: red, green, and blue. Let’s say that the spinner is equally likely to land on one of the three colors.

It is only 1/3 likely that it will land on red if it is spun once.

Let’s say that the spinner is spun 100 times and it lands twice on red, once on the green, and once again on blue. It is unlikely that it will land on any one of these colors.

We assume that P(red), the probability of the spinner landing in red on a given spin, is 1/3.

When calculating the likelihood, however, we are trying to determine if model parameters (P(red), P(green), and P(blue), are correctly specified.

The 100 spins above make it highly unlikely that any color will occur equally in the above example.

Example 3: Probability vs. Likelihood in Gambling

Imagine a casino claiming that 40% of the chances of winning on a particular slot machine are possible for each turn.

The probability of winning money if we only take one turn is 0.40.

Let’s say we win 42 times in 100 turns. It seems fair to conclude that 40% of the turns are likely to win.

We assume that P(winning), =0.40 when calculating the probability to win on a given turning.

When calculating the likelihood, however, we are trying to determine if P(winning) = 0.40 was correctly specified.

The example above shows that winning 42 times out of 100 is a reasonable probability.