Answer

Verified

413.7k+ views

**Hint:**Use the concept of basic definition of probability and the rule of AND and OR between the events.

$P(Event)=\dfrac{\rm{Favourable \space cases}}{\rm{Total \space cases}}$

\[P\left( A\text{ }or\text{ }B \right)\text{ }=\text{ }P\left( A \right)\text{ }+\text{ }P\left( B \right)\]

\[P\left( A\text{ }and\text{ }B \right)\text{ }=\text{ }P\left( A \right)\text{ }\times \text{ }P\left( B \right)\]

We need to find the probability of an article selected at random from the new lot being defective.

**Complete step by step answer:**

Given: There are two lots of identical articles with different amounts of standard and defective articles.

$N$ article in the first lot, $n$ of which are defective

$M$ articles in the second lot, $m$ of which are defective.

$K$ articles are selected from the first lot and $L$ articles from the second and a new lot of results.

$P({{E}_{1}})$: The selected article is from ${{1}^{st}}$ lot $=\dfrac{\rm{Number\text{ }of\text{ }articles\text{ }selected\text{ }from\text{ }the\text{ }first\text{ } lot}}{\rm{Total\text{ }article\operatorname{s}\text{ }in\text{ }the\text{ }new\text{ } lot\text{ }formed}}$ $=\dfrac{K}{K+L}$

$P({{E}_{2}})$ : The selected article is from ${{2}^{nd}}$ lot $=\dfrac{\rm{Number\text{ }of\text{ }articles\text{ }selected\text{ }from\text{ }the\text{ }\sec ond\text{ } lot}}{\rm{Total\text{ }article\operatorname{s}\text{ }in\text{ }the\text{ }new\text{ } lot\text{ }formed}}$ $=\dfrac{L}{K+L}$

Required Probability:\[\begin{align}

& ={\rm(Particle\text{ }selected\text{ }at\text{ }random\text{ }from\text{ }the\text{ }new\text{ } lot\text{ }being\text{ }defective)} \\

& ={\rm(Particle\text{ }selected\text{ }from\text{ }the\text{ }first\text{ } lot\text{ }being\text{ }defective)\text{ }} or \text{ } {\rm(Particle\text{ }selected\text{ }from\text{ }the\text{ }\sec ond\text{ } lot\text{ }being\text{ }defective)} \\

\end{align}\]

$\begin{align}

& =\operatorname{\rm{P}(selected\text{ }articl{{e}_{{{1}^{st}}}} lot)\text{ }} {\rm and} \operatorname{\rm{P}(defective\text{ }articl{{e}_{{{1}^{st}}}} lot)}+\operatorname{\rm{P}(selected\text{ }articl{{e}_{{{2}^{nd}}}} lot)\text{ }} {\rm and}\operatorname{\rm{P}(defective\text{ }articl{{e}_{{{2}^{nd}}}} lot)} \\

& =\operatorname{\rm{P}(selected\text{ }articl{{e}_{{{1}^{st}}}} lot)}\times \operatorname{\rm{P}(defective\text{ }articl{{e}_{{{1}^{st}}}} lot)}+\operatorname{\rm{P}(selected\text{ }articl{{e}_{{{2}^{nd}}}} lot)}\times \operatorname{\rm{P}(defective\text{ }articl{{e}_{{{2}^{nd}}}} lot)} \\

& =\left( \dfrac{K}{K+L} \right)\times \dfrac{n}{N}+\left( \dfrac{L}{K+L} \right)\times \dfrac{m}{M} \\

& =\dfrac{1}{K+L}\left( \dfrac{Kn}{N}+\dfrac{Lm}{M} \right) \\

& =\dfrac{LmN+KnM}{NM(K+L)} \\

\end{align}$

**Hence the correct answer is Option A.**

**Note:**In such type of questions which involves probability knowing the definition of probability combined with OR and AND rule is needed. Accordingly follow the steps to find the required answer.

Recently Updated Pages

The base of a right prism is a pentagon whose sides class 10 maths CBSE

A die is thrown Find the probability that the number class 10 maths CBSE

A mans age is six times the age of his son In six years class 10 maths CBSE

A started a business with Rs 21000 and is joined afterwards class 10 maths CBSE

Aasifbhai bought a refrigerator at Rs 10000 After some class 10 maths CBSE

Give a brief history of the mathematician Pythagoras class 10 maths CBSE

Trending doubts

Difference Between Plant Cell and Animal Cell

Give 10 examples for herbs , shrubs , climbers , creepers

Name 10 Living and Non living things class 9 biology CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

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

Fill the blanks with proper collective nouns 1 A of class 10 english CBSE

Write the 6 fundamental rights of India and explain in detail