# State and prove the Addition theorem on probability.

Answer

Verified

174.3k+ views

**Hint:**Since this is a question related to probability, so there is occurrence of events say two events and this theorem involves addition on them. Probability of an event is the number of ways events can occur divided by the total number of possible outcomes.

**Complete step-by-step answer:**

Statement of the addition theorem on probability:

If A and B are any two events of a random experiment and P is a probability function then the probability of happening of at least one of the events is defined as $P\left( {A \cup B} \right) = P\left( A \right) + P\left( B \right) - P\left( {A \cap B} \right)$.

Now, we have to prove the Addition theorem of probability.

Given: A and B are any two events of a random experiment.

To prove: $P\left( {A \cup B} \right) = P\left( A \right) + P\left( B \right) - P\left( {A \cap B} \right)$.

Proof:

From the set theory we know that

$n\left( {A \cup B} \right) = n\left( A \right) + n\left( B \right) - n\left( {A \cap B} \right)$.

Suppose $n\left( S \right)$ denote the total number of the possible events of random experiment and then dividing both left hand side and right hand side of the of the above equation we get,

\[\dfrac{{P\left( {A \cup B} \right)}}{{n\left( S \right)}} = \dfrac{{P\left( A \right)}}{{n\left( S \right)}} + \dfrac{{P\left( B \right)}}{{n\left( S \right)}} - \dfrac{{P\left( {A \cap B} \right)}}{{n\left( S \right)}}\]

Now, we know that a formula for probability $P\left( x \right) = \dfrac{{n\left( x \right)}}{{n\left( S \right)}}$. By applying this we can write

\[P\left( {A \cup B} \right) = P\left( A \right) + P\left( B \right) - P\left( {A \cap B} \right)\].

**Hence, the above given addition theorem of probability is proved.**

**Note:**

Special case: If two events A and B are mutually exclusive, then $A \cap B$ is a null set. That is $n\left( {A \cap B} \right) = 0$ So, the probability of happening of at least one of the events is equal to the probability of happening of event A and the probability of happening of event B. Mathematically it is written as $P\left( {A \cup B} \right) = P\left( A \right) + P\left( B \right)$. \[\] Two events A and B are independent events if the equation \[P\left( {A \cap B} \right) = P\left( A \right) \times P\left( B \right)\] holds true.

Recently Updated Pages

The sum of 1 + dfrac25 + dfrac352 + dfrac453 + up to class 11 maths CBSE

If a2 + b2 1 then dfrac1 + b + ia1 + b ia is equal class 11 maths CBSE

A bag contains a white and b black balls Two players class 11 maths CBSE

The chance of India winning the toss is dfrac34 If class 11 maths CBSE

To fill 12 vacancies there are 25 candidates of which class 11 maths CBSE

The line xc cuts the triangle with corners left 00 class 11 maths CBSE

Trending doubts

Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write the 6 fundamental rights of India and explain in detail

Write a letter to the principal requesting him to grant class 10 english CBSE

List out three methods of soil conservation

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

Write a letter to the Principal of your school to plead class 10 english CBSE