A problem on Probabilities

Question

I recently stumbled upon a question on Probabilities. Even though I solved the problem but on second glance , series of questions popped into my mind. The question goes as follows:-

$A$ and $B$ are two candidates seeking admission in a college. The probability that A is selected is 0.5, probability that both $A$ and $B$ are selected is 0.3 . What's the maximum probability that $B$ is selected ?

The usual solution goes as follows:-

Let $P(A)$ be the probability that $A$ is selected , and $P(B)$ be the probability that $B$ is selected. Then we have:- $$P(A \cup B) \le 1$$ $$P(A) + P(B) - P (A \cap B)\le 1$$ $$0.5 + P(B) - 0.3 \le 1$$ $$P(B) \le 0.8$$

I interpreted the question in this way:- Consider that mock selections are conducted 10 times.According to the given data $A$ is selected in 5 of them. The cases when both are selected are at most 3. So there must be 3 mock selections common to both these candidates. Therefore the minimum number of selections that $B$ can enjoy is 3 . There are 5 other mock selections in which $A$ is not selected. Favouring maximum selections,$B$ can be selected in all of them. All in all, $B$ is selected in 8 of these mock selections which gives us the required probability.

My query is , Are these two events dependent on each other? Does selection of one candidate reduce the chances of selection of the other candidate? If these events are independent can I use $P(A)*P(B)=P(A \cap B)$ giving me 0.6 as the answer? In the usual solution does the approach consider all cases of dependency or independency of events (Consider the case below)?

Assuming that the above approach considers all cases which include independent events too let's consider two events $x$ and $y$. If both are independent their intersection would be equal to $x*y$. Now consider this. If independent events are a subset of events of occurrence of both then:- $$ x*y \le x\cap y$$ $$ x*y \le 1-x-y$$ $$(1+x)(1+y) \le 2$$ Which is only true for certain values of x and y. ( You can't plug in values like 0.9 and 0.1)? Where am I going wrong? Any help would be appreciated. Thanks in advance.

Answer 1

As to whether the events are independent, there's no way to tell. Suppose that there are a fixed number of positions in the entering class. Then the admission of any candidate might reduce the chances of admission of another candidate, so the two events are dependent. On the other hand, the college may divide the entering classes into different groups in quest of diversity, perhaps, and select a predetermined number of candidate from each group. The selection of two candidates from different groups might well be independent.

Answer 2

The possibility that both $A$ and $B$ are selected depends on the possibility of each of the two candidates are selected, in this case equals to the product of the possibility that $A$ are selected and the possibility that $B$ is selected.

The possibility that $B$ is selected and the possibility that $A$ is selected depend on each other, but only if the possibility of both $A$ and $B$ get selected is a constant (in this case it's $0.3$).

Let $x$ be the possibilty of $B$ is selected, we have this easy equation:

$0.5x=0.3$, so $x=0.6$.

The possibility of $B$ is selected is always $0.6$.