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 E_{1}, the probability of E_{1} It is done with P (E_{1})

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 E_{1 } be a simple event and let A be a composite event.

0 ≤ P (E_{1}) 1

0 ≤ P (A) ≤ 1

**Feature # 2:**

The sum of all simple events is equal to 1

Let E_{1 , }E2_{, … }be simple events of an experiment. Then P (E_{1}) + 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 =

1

4th

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

Let E_{1} be a simple event and let A be a composite event.

In general,

SPORTS_{1}) =

1

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 =

50

1000

= 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 =

80

200

= 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.