$A$ is a set containing $n$ elements,a subset $P$(may be void also) is selected at random from set $A$ and the set $A$ is then reconstructed by replacing the elements of $P.$A subset $Q$(may be void also)of $A$ is again chosen at random.
$(A)$What is the probability that number of elements in $P$ is equal to the number of elements in $Q?$
$(B)$What is the probability that $P\cap Q=\emptyset?$
I could not solve this question,i have no idea how to even start it.Please help me.
Best Answer
First problem; It is convenient to imagine that the choices are made from two distinct $n$-element sets, $A$ and $B$, say boys and girls. There are $2^n$ equally likely ways to choose a subset of $A$, and $2^n$ equally likely ways to choose a subset of $B$, for a total of $2^{2n}$.
There are $\binom{n}{k}$ ways to choose a $k$-subset of $A$, and $\binom{n}{k}$ ways to choose a $k$-subset of $B$, for a total of $\binom{n}{k}^2$ ways. Thus the total number of "favourables" is $\sum_{k=0}^n \binom{n}{k}^2$.
Now we have our answer, but it can be greatly simplified. What follows has appeared a number of times on MSE. Replace one of the $\binom{n}{k}$ by $\binom{n}{n-k}$. So we are interested in the sum $\sum_0^n \binom{n}{k}\binom{n}{n-k}$.
But this sum is the number of ways to choose a total of $n$ people from the $n$ boys and $n$ girls, so the number of favourables is $\binom{2n}{n}$. Divide by $2^{2n}$ for the probability.
Second problem: Imagine that we are choosing the two sets $P$ and $Q$ by writing down the letter P if the point is to be in $P$, writing NP if the point is to be not in $P$, and doing something similar for $Q$. Then we are asking for the probability we do not write down simultaneously P and Q. The probability of this for any element of $A$ is $\frac{3}{4}$, so the probability this happens $n$ times in a row is $\left(\frac{3}{4}\right)^n$.