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Mathematics Question on Conditional Probability

A bag contains nn balls. It is given that the probability that among these n balls exactly rr balls are white is proportional to r2(0≤r≤n)r^2 (0 \le r \le n). A ball is drawn at random and is found to be white. Then the probability that all the balls in the bag are white, will be:

A

2n(n+1)2\frac{2n}{\left(n+1\right)^{2}}

B

4n(n+1)2\frac{4n}{\left(n+1\right)^{2}}

C

2n(n+3)2\frac{2n}{\left(n+3\right)^{2}}

D

4n(n+3)2\frac{4n}{\left(n+3\right)^{2}}

Answer

4n(n+1)2\frac{4n}{\left(n+1\right)^{2}}

Explanation

Solution

Let Ei=0,1,2,...nE_i = 0, 1, 2, ... n be the event that the bag contains exactly i white balls then p(Ei)k2ip (E_i) k^2i where ∑i=0nki2=1\displaystyle \sum_{i=0}^n k^2_i=1 ⇒ k=rn(n+1)(2n+1)\Rightarrow \:k=\frac{r}{n\left(n+1\right)\left(2n+1\right)} Let A be the event that a ball drawn is white p(EnA)=p(En)p(A/En)∑i=0nki2(in)p\left(\frac{E_n}{A}\right)=\frac{p\left(E_n\right)p\left(A/E_n\right)}{\displaystyle \sum_{i=0}^nk^2_i\left(\frac{i}{n}\right)} =K.n2.1kn∑t=0n(i)3=\frac{K.n^2.1}{\frac{k}{n}\displaystyle \sum_{t=0}^n(i)^3} =n3(n(n+1)2)2=\frac{n^{3}}{\left(\frac{n\left(n+1\right)}{2}\right)^{2}} p(EnA)=4n(n+1)2p\left(\frac{E_{n}}{A}\right)=\frac{4n}{\left(n+1\right)^{2}}