A lot of 100 watches are known to have 10 defective watches. If 8 watches are selected (one by one with replacement) at random, the probability that there are at least one defective watch is ${\text{ = 1 - }}{\left( {\dfrac{9}{{10}}} \right)^x}$, then the value of x is ………………………………….
Last updated date: 16th Mar 2023
•
Total views: 308.1k
•
Views today: 5.87k
Answer
308.1k+ views
Hint – Use probability distribution of random variables. Probability distribution provides the possibility of presence of different outputs.
In a lot of 100 watches we know that 10 are defective.
We have to select 8 watches one by one without replacement.
Let X denote the number of defective watches in 8 draws and let P be the probability of selecting a defective watch in a draw.
Now, X follows binomial distribution with parameters ${\text{n = 8}}$ and ${\text{p = }}\dfrac{{10}}{{100}} = \dfrac{1}{{10}}$ because we have total 100 watches out of which 10 are defective.
Now ${\text{P}}\left( {X = r} \right) = {}^n{c_r}{\left( p \right)^r}{\left( {1 - p} \right)^{n - r}}$
Using the above concept
$P\left( {X = r} \right) = {}^8{c_r}{\left( {\dfrac{1}{{10}}} \right)^r}{\left( {\dfrac{9}{{10}}} \right)^{8 - r}}$Where our ${\text{where our r = 0,1,2}}....{\text{8}}$
Now we are asked to find the probability that at least one defective watch is drawn.
So we have to find ${\text{P}}\left( {X > = 1} \right)$
Now ${\text{P}}\left( {X > = 1} \right) = 1 - P\left( {X = 0} \right)$
This is equal to ${\text{1 - }}{}^8{c_0}{\left( {\dfrac{1}{{10}}} \right)^0}{\left( {\dfrac{9}{8}} \right)^8} = 1 - {\left( {\dfrac{9}{8}} \right)^8}$
Hence the value of required ${\text{x = 8}}$
Note –Whenever we face such a type of problem statement the key concept that we need to recall is the concept of probability distribution of random variables .This helps to solve such a type of question and it will get you on the right track to reach the answer.
In a lot of 100 watches we know that 10 are defective.
We have to select 8 watches one by one without replacement.
Let X denote the number of defective watches in 8 draws and let P be the probability of selecting a defective watch in a draw.
Now, X follows binomial distribution with parameters ${\text{n = 8}}$ and ${\text{p = }}\dfrac{{10}}{{100}} = \dfrac{1}{{10}}$ because we have total 100 watches out of which 10 are defective.
Now ${\text{P}}\left( {X = r} \right) = {}^n{c_r}{\left( p \right)^r}{\left( {1 - p} \right)^{n - r}}$
Using the above concept
$P\left( {X = r} \right) = {}^8{c_r}{\left( {\dfrac{1}{{10}}} \right)^r}{\left( {\dfrac{9}{{10}}} \right)^{8 - r}}$Where our ${\text{where our r = 0,1,2}}....{\text{8}}$
Now we are asked to find the probability that at least one defective watch is drawn.
So we have to find ${\text{P}}\left( {X > = 1} \right)$
Now ${\text{P}}\left( {X > = 1} \right) = 1 - P\left( {X = 0} \right)$
This is equal to ${\text{1 - }}{}^8{c_0}{\left( {\dfrac{1}{{10}}} \right)^0}{\left( {\dfrac{9}{8}} \right)^8} = 1 - {\left( {\dfrac{9}{8}} \right)^8}$
Hence the value of required ${\text{x = 8}}$
Note –Whenever we face such a type of problem statement the key concept that we need to recall is the concept of probability distribution of random variables .This helps to solve such a type of question and it will get you on the right track to reach the answer.
Recently Updated Pages
If ab and c are unit vectors then left ab2 right+bc2+ca2 class 12 maths JEE_Main

A rod AB of length 4 units moves horizontally when class 11 maths JEE_Main

Evaluate the value of intlimits0pi cos 3xdx A 0 B 1 class 12 maths JEE_Main

Which of the following is correct 1 nleft S cup T right class 10 maths JEE_Main

What is the area of the triangle with vertices Aleft class 11 maths JEE_Main

KCN reacts readily to give a cyanide with A Ethyl alcohol class 12 chemistry JEE_Main

Trending doubts
What was the capital of Kanishka A Mathura B Purushapura class 7 social studies CBSE

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

Tropic of Cancer passes through how many states? Name them.

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

Name the Largest and the Smallest Cell in the Human Body ?
