For independent events $A$ and $B$,$P(A) = \frac{1}{3}$,$P(A \cup B) = \frac{2}{5}$,and $P(B) = p$. Find the value of $p$.

$A$ woman has $10$ keys,out of which only one opens a lock. She tries the keys one after another (keeping aside the failed ones) until she succeeds in opening the lock. What is the probability that the seventh key works?

If $P(A) = \frac{3}{5}$ and $P(B) = \frac{1}{5}$,find $P(A \cap B)$ if $A$ and $B$ are independent events.

From a pack of $52$ cards,two cards are drawn in succession one by one without replacement. The probability that both are aces is

Two cards are drawn one by one at random from a pack of $52$ cards. The probability that both of them are king,is

$A$ box of oranges is inspected by examining three randomly selected oranges drawn without replacement. If all the three oranges are good,the box is approved for sale; otherwise,it is rejected. Find the probability that a box containing $15$ oranges,out of which $12$ are good and $3$ are bad,will be approved for sale.