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 $p$.

Let $A$ and $B$ be independent events with $P(A)=0.3$ and $P(B)=0.4$. Find $P(A \cap B)$.

Bag $B_1$ contains $4$ white and $2$ black balls. Bag $B_2$ contains $3$ white and $4$ black balls. $A$ bag is chosen at random and a ball is drawn from it at random,then the probability that the ball drawn is white,is

Two balls are drawn at random with replacement from a box containing $10$ black and $8$ red balls. Find the probability that the first ball is black and the second is red.

For independent events $A$ and $B$,if $P(A) = p$,$P(B) = \frac{1}{2}$,and $P(A \cup B) = \frac{3}{5}$,then 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?