What is probability?

What is probability? How to calculate probability? We will answer these questions here along with some useful properties of probability.

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

For a simple event E1, the likelihood of E1 happening is denoted by P(E1)

For a compound event A, the likelihood of A happening is denoted by P(A)

Conceptual approaches to probability / How to calculate probability

There are 3 ways to approach probability as shown below.

  • Classical probability
  • Relative frequency of probability
  • Subjective probability.


Classical probability 

Classical probability is used for equally likely outcomes.

Equally likely outcomes have the same probability of occurrence. For example, most people know that if you toss a coin, it is 50/50 chance.

All outcomes are equally likely since neither head nor tail has a better chance of occurring.

Head can occur 50% of the time and tail can occur 50% of the time as well.

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

The outcomes are still equally likely. This time though each outcome can occur 25% of the time.

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

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

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

We then have the following 2 properties:

Property #1:

The probability of an event is always between 0 and 1

Let E be a simple event and let A be a compound event.

0 ≤ P(E1) ≤ 1

0 ≤ P(A) ≤ 1

Property #2:

The sum of all simple events is equal to 1

Let E1 ,  E2, ... be simple events of an experiment. Then, P(E1) + P(E2) + P(E3)+ ... = 1

Classical probability formula

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

Notice also that 0.25 =  
1 / 4

The 1 in the numerator represents the sum of all simple events and the 4 is the number of outcomes.

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

In general,

P(E1) =  
1 / Total number of outcomes for the experiment

P(A) =  
Number of outcomes favorable to A / Total number of outcomes for the experiment


Relative frequency of probability

Suppose you want to calculate the following probabilities

  • The probability that a randomly selected family owns 2 cars
  • The probability that a loaded coin will yield tail

These two experiments will not yield equally likely outcomes.

Recall that when you flip a coin once, you either get head or tail.

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

If the coin is loaded, it will no longer be 50/50.

The same goes for these two outcomes, " A family owns 2 cars " and " A family does not own 2 cars "

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

In reality, this will never be the case, so classical probability cannot be used to compute this probability.

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

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

P(A) =  
f / n

To compute the probability that a randomly selected family owns 2 cars, you may need to conduct a survey.

Take a random sample of 1000 people for instance and ask them how many cars they own. If 50 of them say they own 2 cars, you can use the formula to find the probability.

Probability that a family owns 2 cars =
50 / 1000
= 0.05 = 5%

For the coin, you may want to toss the coin a great number of times and see how many times it yields tail. Say you throw it 200 times and it yields tail 60 times.

Probability that the coin will yield tail =
80 / 200
= 0.4 = 40%

Subjective probability

Some experiments neither have equally likely outcomes nor can be repeated to generate data.

Some examples of subjective probability includes

  • The probability that the US men's soccer team will win the next world cup.
  • The probability that the minimum wage will increase to 15 dollars in the US.
  • The probability that a student will get an A in a math class.

To find the subjective probability of these events above, you will need to make an educated guess using judgment, experience, belief, or information available on the event.

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