Question

A bag containing 7 red, 5 black and 8 black balls. If four balls are drawn one by one with replacement, the probability that all are white$\dfrac{1}{{{{16}^x}}}$, what is the value of x?

Answer 1

Hint: Here we need to have knowledge about the probability concept and basic idea about the random selection process which will be helpful to solve the problem.

According to the data given,
In total we have 20 balls, which includes 7 red balls, 5black balls and 8 black balls.
Let us consider p is the probability of getting all white from 7 red balls, 5black balls and 8 black balls.
So here we have 20 balls in total and 5 white balls
Therefore we can say that the probability of getting all white balls from total number of balls is
$ \Rightarrow p = \dfrac{5}{{20}} = \dfrac{1}{4}$
And we also know that p is complement of q
So, p=q-1
$
   \Rightarrow q = 1 - \dfrac{1}{4} \\
   \Rightarrow q = \dfrac{3}{4} \\
$

Now let us consider X which denotes the random variable of the number of selecting white balls with the replacement out of 4 balls.
Therefore the probability of getting r white balls out of n is can be given as
$p({\rm X} = r){ = ^n}{C_r}{(p)^r}{(q)^{n - r}}$
As we know the p and q values let us substitute the value to get the probability of white balls
$ \Rightarrow p({\rm X} = r){ = ^n}{C_r}{\left( {\dfrac{1}{4}} \right)^r}{\left( {\dfrac{3}{4}} \right)^{n - r}}$
So here we have to find the probability of getting all white balls is

Therefore the probability of getting all white balls $ = p({\rm X} = 4){ = ^4}{C_4}{\left( {\dfrac{1}{4}} \right)^4}{\left( {\dfrac{3}{4}} \right)^{4 - 0}} = \dfrac{1}{{{4^4}}} = \dfrac{1}{{{{16}^2}}}$

Hence we they mentioned that If four balls are drawn one by one with replacement, the probability that all are white$\dfrac{1}{{{{16}^x}}}$
Now on comparing the given values with result, we say that x value is 2
Therefore x=2

NOTE: In this problem they have already mentioned the probability of getting all white balls is equal to some value to the power x. So here we have to find the x value. For this first we have to find the probability of all white balls within the total number of balls given later on using the probability values we have to find the probability of all balls in the random variable case to get the x value.