Independent Events and Probability

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Independent events and probability can be defined as those occurrences that are not dependent on any specific event. A good example will be if an individual flips a coin, then he/she has the chance of getting head or tail. In both outcomes, the occurrences are independent of each other, which makes an event of probability. This theory can be understood with the Venn diagram, which gives equally exclusive events.

In this section, students will learn independent events probability formula and its application in the equation. These independent and dependent events require trials and circumstances to justify the explanation. 

To understand the concepts better, let’s dive into the independent event definition.

What is an Independent Events Probability?

In an independent event and probability, the outcomes in an experiment are termed as events. Ideally, there are multiple events like mutually exclusive events, independent events, dependent events and more.

In a situation, if other event C’s occurrence doesn’t harm a probability of occurrence of an event B, then B and C are called independent events.

Let’s take an example where an individual rolls a dice. Here an event D is the appearance odd numbers and E is the multiple of number 3 in an event. The situation will be 

P(D)= 3/6 = 1/2 and P(E) = 2/6 = 1/3

It is to note here that D and E are the odd number outcome and multiples of 3 respectively. So, P (D ∩ E) = 1/6

P(D│E) = P (D ∩ E)/ P(E)

This makes a total of 1613 which is equal to 12

Here P(D) = P(D│E) = 1/2, which entails that the occurrence of event E will not be affected by the probability of event D’s occurrence.

Taking A and B as independent events, then P(D│E) = P(D)

After applying the multiplication rule of probability, we get (D ∩ E) = P(D) 

This makes P(D│E)

P (D ∩ E) = P(E) 


What Does the Probability of Dependent Events Mean?

To understand the probability of dependent events, young learners need to make their concept of probability clear. The measure of the likelihood of an event can be termed as a probability. It is impossible to predict every event happening with certainty. An individual can guess the chances of occurrence which is based on judgement. 

Ideally, a probability ranges from 0 to 1 where zero indicates the impossibility of an event while one can define the possibility of an event. Students need to understand the basic concept of dependent events which relies on others. It is seen that when there are two dependent events, one of them influences the probability of others. It ideally depends on the occurrence of another event first. 

It is essential to know that probability of almost every event in a trial space adds up to 1.

Let’s take an example -

If B and C are dependent events, then the probability of C and the probability of B happening will be 

P(B) × P (C after B) when B is given

Now P (B and C) = P(B) × P (C after B)

 P(C after B) can be written as P(C | B)

 This makes P (B and C) = P(B) × P (C | B)

Apart from practising equations based on Independent events and probability, a student needs proper guidance on related theorems. For strengthening the base on mathematical equations, they require top-notch study materials and test papers.

One can check Vedantu, which is a trustworthy education site offering solutions on mutually exclusive and independent events and more. Apart from test papers, they offer pocket-friendly live classes and notes based on the probability of two independent events.

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FAQ (Frequently Asked Questions)

1. What is the Value of P (A and B), P (A or B), P (A not B), and P (neither A nor B) if X and Y are Two Independent Events or Independence of Events Such That P(A) = 0.4 and P(B) = 0.8?

Ans. Given P(A) = 0.4 and P(Y) = 0.8 and events A and B are independent of each other.

P (A and B) = P (A ∩ B) = P(A) P(B) = 0.4 × 0.8 = 0.32

P (A or B) = P (A ∪ B) = P(A) + P(B) – P (A ∩ B) = 0.4 + 0.8 – 0.32 = 0.88

P (B not A) = P (B ∩ A’) = P(B) – P (A ∩ B) = 0.8 – 0.32 = 0.48

and P (neither A nor B) = P (A’ ∩ B’) = 1 – P (A ∪ B) = 1 – 0.88 = 0.12

2. What is Experimental Probability?

Ans. Experimental probability is ideally based on ample recording of events and occurrence based on experiments. They are also termed Empirical probability due to their experimental nature. A series of experiments are conducted to find occurrences of events. These are called random as there are no fixed results. These trials run on uncertainty.

Ideally, unsystematic experiments are repeated numerous times to find the likelihood. An experiment is repeated a fixed number of times which is also called a trial. This makes the formula for the experimental probability as P(E) = number of times an event occurs divided by the total number of trials.

3. What is Theoretical Probability? Give an Example of This Theorem.

Ans. Theoretical probability can be termed as a theory predicting the reason behind probability. To find the reason behind an event one doesn’t require experimenting when using theoretical probability. Instead, one must use the variables of the situation to find the occurrence of the probability of an event. 

The theoretical probability can also be termed as the ratio of the number of possible outcomes to the number of favourable outcomes. 

Therefore, a probability of event or P(E) is the number of favourable outcomes divided by the number of possible outcomes.

Example- An individual desiring to find the probability of 4 when rolling dice doesn’t require experiment. There are ideally six outcomes when one rolls a dice which are 1, 2, 3, 4, 5. 

Therefore, a probability of event or P(E) is the number of favourable outcomes divided by the number of possible outcomes.

Applying the formula P(E) = 1/6.

Hence, the probability of getting 4 while rolling dice is 1/6.