Answer
424.2k+ views
Hint: Mutually exclusive events are the events that do not occur at the same time occur in a statistical experiment. Here we can see that the outcome of one event does not depend upon another event. Hence the probability of occurring one event is also independent of occurring of other events. Try to use this fact to solve the problem.\[\]
Complete step-by-step answer:
Let us denote $ A $ as the set containing the defective items and $ B $ as the set containing non-defective items. An item can either be defective or non-defective. Hence $ A\text{ and }B $ are disjoint sets and the sample space is the set \[S=A\bigcup B \]
Now if randomly chosen, the probability of an item being defective
\[P\left( A \right)=\dfrac{n(A)}{n(S)}=\dfrac{7}{35}\left( \text{where }n\text{ denotes the number of items in the set}\text{.} \right)\]
Similarly the probability of an item being non-defective is
\[P\left( B \right)=\dfrac{n(B)}{n(S)}\]
Total Probability or the probability of sample space is
\[P(S)=P\left( A\bigcup B \right)=\dfrac{n(A\bigcup B)}{n(S)}=\dfrac{n(S)}{n(S)}=1\]
As $ A\text{ and }B $ are disjoint sets then $ A\bigcap B=\varnothing $ .
The probability both the events occurring at the same time is given by \[P\left( A\bigcap B \right)=\dfrac{n(A\bigcap B)}{n(S)}=\dfrac{0}{n(S)}=0\]
From the general formula of probability,
\[\begin{align}
& P\left( A\bigcup B \right)=P\left( A \right)+P\left( B \right)-P\left( A\bigcap B \right) \\
& \Rightarrow 1=\dfrac{7}{35}+P\left( B \right) \\
& \Rightarrow P\left( B \right)=1-\dfrac{7}{35}=\dfrac{28}{35} \\
\end{align}\]
The probability is found to be $ \dfrac{4}{5} $ .\[\]
Note: The question checks whether you can differentiate mutually exclusive events and mutually inclusive events. Mutually inclusive are events that occur at the same time. You can use the following formula for $ n $ such mutually exclusive events to find out probabilities. \[P\left( \bigcup\limits_{i=1}^{n}{{{A}_{i}}} \right)=P\left( {{A}_{1}} \right)+P\left( {{A}_{2}} \right)+P\left( {{A}_{3}} \right)...+P\left( {{A}_{n}} \right)\] Please note that mutual exclusiveness or inclusiveness of probabilistic events is not necessarily related to dependent and independent events.
Complete step-by-step answer:
Let us denote $ A $ as the set containing the defective items and $ B $ as the set containing non-defective items. An item can either be defective or non-defective. Hence $ A\text{ and }B $ are disjoint sets and the sample space is the set \[S=A\bigcup B \]
Now if randomly chosen, the probability of an item being defective
\[P\left( A \right)=\dfrac{n(A)}{n(S)}=\dfrac{7}{35}\left( \text{where }n\text{ denotes the number of items in the set}\text{.} \right)\]
Similarly the probability of an item being non-defective is
\[P\left( B \right)=\dfrac{n(B)}{n(S)}\]
Total Probability or the probability of sample space is
\[P(S)=P\left( A\bigcup B \right)=\dfrac{n(A\bigcup B)}{n(S)}=\dfrac{n(S)}{n(S)}=1\]
As $ A\text{ and }B $ are disjoint sets then $ A\bigcap B=\varnothing $ .
The probability both the events occurring at the same time is given by \[P\left( A\bigcap B \right)=\dfrac{n(A\bigcap B)}{n(S)}=\dfrac{0}{n(S)}=0\]
From the general formula of probability,
\[\begin{align}
& P\left( A\bigcup B \right)=P\left( A \right)+P\left( B \right)-P\left( A\bigcap B \right) \\
& \Rightarrow 1=\dfrac{7}{35}+P\left( B \right) \\
& \Rightarrow P\left( B \right)=1-\dfrac{7}{35}=\dfrac{28}{35} \\
\end{align}\]
The probability is found to be $ \dfrac{4}{5} $ .\[\]
Note: The question checks whether you can differentiate mutually exclusive events and mutually inclusive events. Mutually inclusive are events that occur at the same time. You can use the following formula for $ n $ such mutually exclusive events to find out probabilities. \[P\left( \bigcup\limits_{i=1}^{n}{{{A}_{i}}} \right)=P\left( {{A}_{1}} \right)+P\left( {{A}_{2}} \right)+P\left( {{A}_{3}} \right)...+P\left( {{A}_{n}} \right)\] Please note that mutual exclusiveness or inclusiveness of probabilistic events is not necessarily related to dependent and independent events.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Why Are Noble Gases NonReactive class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let X and Y be the sets of all positive divisors of class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
At which age domestication of animals started A Neolithic class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Write a letter to the principal requesting him to grant class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)