Question

An urn contains 4 white and 3 red balls. Three balls are drawn with replacement from this urn. Then, the standard deviation of the number of red ball drawn is
A.\[\dfrac{6}{7}\]
B.\[\dfrac{36}{49}\]
C.\[\dfrac{5}{7}\]
D.\[\dfrac{25}{49}\]

Answer 1

Hint: The variance of \[X\] is the arithmetic mean of the squares of all deviations of \[X\] from the arithmetic mean of the observations and it is denoted by \[Var(X)\] or \[\sigma _{{}}^{2}\]. The position square root of the variance of \[X\] is known as the standard deviation and it is denoted by \[\sigma \].
Standard deviation \[=\dfrac{6}{7}\]
To find the standard deviation first variance has to be found. Variance can be calculated using the formula given below
\[Var(X)=\dfrac{1}{n}\sum\limits_{i=1}^{n}{x_{i}^{2}}-{{\left( \dfrac{1}{n}\sum\limits_{i=1}^{n}{{{x}_{i}}} \right)}^{2}}\]

Complete step-by-step answer:
Let us assume that \[X\]is the number of red balls.
The probability of getting red ball is \[\dfrac{3}{7}\] and the probability of getting white ball is \[\dfrac{4}{7}\]
Using the Binomial distribution concept to solve
Here \[n\] is the number of red balls and \[p\] is the probability of getting the red balls and \[q\] is the probability of not getting the red balls.
By observing we get
\[n=3\]
\[p=\dfrac{3}{7}\]
And \[q=\dfrac{4}{7}\]
In Binomial distribution,
\[Mean=np\]
\[Variance=npq\]
And Standard Deviation \[=\sqrt{npq}\]
So after applying the values we get
\[Mean=3\times \dfrac{3}{7}\]
Further solving we get the mean as
\[Mean=\dfrac{9}{7}\]
For variance apply the value in the formula
\[Variance=3\times \dfrac{3}{7}\times \dfrac{4}{7}\]
Further solving we get the variance as
\[Variance=\dfrac{36}{49}\]
For Standard Deviation use the formula \[\sqrt{npq}\]
Substituting the values we get
Standard Deviation \[=\sqrt{\dfrac{36}{49}}\]
Further solving and simplifying we get
Standard Deviation \[=\dfrac{6}{7}\]
Therefore, the standard deviation is\[\dfrac{6}{7}\].
So, the correct answer is “Option A”.

Note: The mean deviation can also be used for calculation of variance and standard deviation. Compute the mean of the given observation and take the deviations of the observations from the mean. Square the deviation obtained and obtains the sum. Then, divide the sum. This will give the variance value.