Probability is a mathematics discipline connected with the numerical description of the likelihood of an event to occur. The probability of an event occurring can only be between 0 and 1. 0 shows the impossibility of the event, and 1 shows the certainty of happening.

For example, when tossing a coin, there is an equal probability of getting a ”head” or a ”tail”. The likelihood of getting a ”head” or a ”tail’ is 1/2, which is 0.5 or 50%.

**What Is Conditional Probability?**

Conditional probability is the likelihood of an event (X) considering that (Y) has already occurred. It is the probability of an outcome happening considering the existence of a previous outcome.

**Formula**

Conditional probability is mathematically written as P(A/B), read as ”the probability of A, given B”.

The conditional probability formula for the occurrence of B when A has already occurred is

**P(B|A) = P(B ∩ A)/P(A)**

While the probability formula for A when B has already occurred is;

**P(A|B) = P(A ∩ B)/P(B)**

**What is Theoretical Probability?**

Theoretical probability is the theory behind probability. It is not mandatory to experiment when using theoretical probability to find the probability of an event. Theoretical probability is described as the number ratio of the favorable outcomes to the number of possible outcomes.

## **The** **Probability Theory**

Probability theory is the mathematical structure that allows you to scrutinize events logically. You can also call it the probability theory the logic of science. Probability theory examples are as follows;

- Experimental – It is also known as empirical. It defines probability as thought experiments. It was obtained as a result of experiments repeated several times. You can get more help with probability assignments online.
- Theoretical – It is a probability that is determined based on reasoning.
- Subjective – This probability deals with judgment and belief that an event will occur. It is what a person expects to happen according to their judgment or beliefs. A good probability theory example is when a football fan predicts that a certain team will win based on the team’s past wins. Not using a formal mathematical calculation.
- Classic – It is a probability that has equal odds of something taking place.
- Conditional – It measures an event probably occurring given that another event has already occurred.

Several rules have to be followed when it comes to probability. They are as follows;

Rule no 1: The probability of an event not happening at all is zero (0), while that of a certain event occurring is one (1). For any event A, 0≤P (A) ≤1

Rule no 2: The total of all the probabilities occurring is equal to one.

Rule no 3: It is the additional rule. It is the probability that either one or both of the events occur.

Rule no 5: It is the probability that both events will occur.

Rule no 6: complement rule. The possibility (probability)of an event not taking place is one minus the probability of it occurring. P (not A)=1-P(A)

**Basic concepts of probability**.

There are three basic concepts in probability;

Chance – It is when events take place in the absence of any intention. It is the possibility of something taking place. You can look at probability math online and work out some of the examples given.

Expectation – It the average value of the random variable where each value is weighted according to its probability. Its value is obtained by multiplying the possible outcomes by the likelihood that each event will occur and then totaling the values.

Variance – It measures how far numerals are spread out from their average value. The advantage of using variance as a measure of dispersion is that it is more flexible to algebraic manipulation than other ways. The disadvantage is that its units are different from the variable.

Variance is a set of observations that are measured from a world that is real. In probability statistics online, the formula of variance is given by squaring each value and multiplying by its probability, totaling them up, and then subtracting the square of the expected value.

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**Probability** **Formula**

**The probability of event P(E) = number of favorable outcomes/numbers of possible outcomes**.

For Example; Find the probability of a 4 in on a dice.

You do not need to experiment to find the probability of getting 4 while tossing a dice since there are only six possible outcomes you can get, i.e., 1,2,3,4,5,6.

P(E) = 1/6

Hence the probability of picking a 4 when tossing a dice is 1/6.

**Types of Probability**

There are various types of probabilities;

### 1. **Classical Probability**

The classical probability states that in a situation where there are A outcomes that are equally likely to happen and event B has specifically C of these outcomes, the probability of B is C/A, or P(B)=C/A. For example, there are only six possible outcomes when rolling dice. So, the probability of rolling each number is 1/6.

### 2. **Empirical Probability**

The empirical probability, also known as experimental probability, explains the likelihood of an outcome through experiments.

For example, suppose you are tossing a weighted die with no idea which side has much weight to have an idea of the possibility of each outcome. In that case, you can toss the die a couple of times while recording the proportion of time a certain outcome is achieved and then estimate the probability of that outcome.

**Probability of event P(E) = The Number of times that event occurs / total number of trials.**

### 3. **Subjective Probability**

Subjective probability is based on own personal judgment and belief that an event will occur. It is the probability that an event that a person expects will occur based on their knowledge without any formal calculations.

For example, investors who trade in stock can predict that the price of the stock will fall at a certain month or a football fan can predict a certain team will win based on their past wins or losses or by analyzing their opponents.

### 4. **Axiomatic Probability**

Axiomatic probability uses axioms or a set of rules. You can quantify the non-occurrence or occurrence of the events in this probability. The possibility of an outcome or event is established through occurrences of previous outcomes or events.

**The Probability Formula**

The probability formula is used to calculate the probability of occurrence of an event.

Probability Formulas

The formula for an event probability is;

**P(A) = Number of favorable outcomes / Total no. of favorable outcomes**

Or

**P(A) = n(A) / n(S)**

Where;

- P(A) is the probability of an outcome or event
- n(A) is the number of favorable outcomes
- n(S) is the sum of events in the sample.

The outcome of interest is the favorable outcome.

**Example;**

What probability will an Ace card be picked from a standard deck?

**Solution.**

The standard pack has 52 cards.

There are 4 Ace cards in a deck of cards

Henceforth, the favorable outcome will be 4

Applying the formula;

**P(Ace) = (no. of favorable outcomes) / (Total no. of favorable outcomes)**

P(Ace) = 4/52 = 1/13

Therefore, the probability of picking an Ace is 1/13.