What is probability How do you calculate the probability? We’ll answer these questions here along with some useful properties of probability.

Probability is a numerical measure of the likelihood that a particular event will occur.

For a simple event E1, the probability of E1 It is done with P (E1)

For a composite event A, the probability that A occurs is denoted by P (A).

Conceptual Approaches to Probability / How to Calculate Probability

There are 3 ways to approach the probability as shown below.

  • Relative frequency of the probability

Classic probability

Classic probability is used for equally likely results.

Equally probable results have the same probability of occurrence. For example, most people know that if you flip a coin the chance is 50/50.

All outcomes are equally likely as neither head nor tail have a better chance of entering.

The head can appear 50% of the time and the tail can also appear 50% of the time.

What if you toss the coin twice? The sample space = {HH, HT, TH, TT}

The results are still equally likely. This time, however, any outcome can appear 25% of the time.

Did you notice that 50% = 0.5 and 25% = 0.25? These values ​​are less than 1. In fact, the greatest value of a probability is 1.

A probability of 1 occurs for events that occur 100% of the time. For events that will never happen, the probability is 0.

Also note that 0.25 + 0.25 + 0.25 + 0.25 = 1 and 0.50 + 0.50 = 1

We then have the following 2 properties:

Feature # 1:

The probability of an event is always between 0 and 1

Let E1 be a simple event and let A be a composite event.

0 ≤ P (E1) 1

0 ≤ P (A) ≤ 1

Feature # 2:

The sum of all simple events is equal to 1

Let E1 , E2, … be simple events of an experiment. Then P (E1) + P (E.2) + P (E.3rd) + … = 1

Classic probability formula

We said earlier that for the sample space s = {HH, HT, TH, TT} the probability of each outcome is 0.25.

Also note that 0.25 =


The 1 in the counter stands for the sum of all simple events and the 4 for the number of results.

Let E1 be a simple event and let A be a composite event.

In general,


Total number of results for the experiment

P (A) =

Number of for A. favorable results
Total number of results for the experiment

Relative frequency of the probability

Suppose you want to calculate the following probabilities

  • The likelihood that a randomly selected family will own 2 cars
  • The likelihood that a charged coin will make a tail

These two experiments will not yield equally likely Results.

Remember that if you flip a coin once you will get either heads or tails.

These two results were equally likely because P (head) = 50% and P (not head) = 50%

When the coin is loaded, it won’t be 50/50 anymore.

The same applies to the two results “A family owns 2 cars” and “A family does not own 2 cars”.

If these two outcomes were equally likely, 50% of the family will own 2 cars and 50% will not own 2 cars.

In reality, this will never be the case, so this probability cannot be calculated using the classical probability.

Instead, we need to use the following formula for relative frequency.

Let A be an event that is observed f times and n the number of repetitions of the experiment.

To calculate the likelihood that a randomly selected family will own 2 cars, you may need to conduct a survey.

For example, take a sample of 1000 people and ask how many cars they own. If 50 of them say they own 2 cars, use the formula to find the probability.

Probability that a family owns 2 cars =


= 0.05 = 5%

For the coin, you might want to flip the coin many times and see how many times it makes a tail. Say you toss it 200 times and it gives tail 60 times.

Probability of the coin making a tail =


= 0.4 = 40%

Subjective probability

Some experiments do not have equally likely results, nor can they be repeated to generate data.

Some examples of subjective probability include

  • The likelihood that the US men’s soccer team will win the next World Cup.
  • The likelihood of the US minimum wage rising to $ 15.
  • The likelihood of a student getting an A in math class.

To determine the subjective likelihood of these above events, you need to make an educated guess based on judgment, experience, belief, or available information about the event.


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