One card is drawn at random from a well shuffled deck of $52$ cards. In which of the following cases are the events $E$ and $F$ independent ?

$\mathrm{E}:$ ' the card drawn is black '

$\mathrm{F}:$ ' the card drawn is a king '

In a deck of $52$ cards, $26$ cards are black and $4$ cards are kings.

$\therefore $ $\mathrm{P}(\mathrm{E})=\mathrm{P}$ (the card drawn is a black ) $=\frac{26}{52}=\frac{1}{2}$

$\therefore $ $\mathrm{P}(\mathrm{F})=\mathrm{P}$ (the card drawn is a king ) $=\frac{4}{52}=\frac{1}{13}$

In the pack of $52$ cards, $2$ cards are black as well as kings.

$\therefore $ $\mathrm{P}(\mathrm{EF})=\mathrm{P}$ (the card drawn is black king ) $=\frac{2}{52}=\frac{1}{26}$

$\mathrm{P}(\mathrm{E}) \times \mathrm{P}(\mathrm{F})=\frac{1}{2} \cdot \frac{1}{13}=\frac{1}{26}=\mathrm{P}(\mathrm{EF})$

Therefore, the given events $\mathrm{E}$ and $\mathrm{F}$ are independent.

The odds against a certain event is $5 : 2$ and the odds in favour of another event is $6 : 5$. If both the events are independent, then the probability that at least one of the events will happen is

Let $A$ and $B$ be two events such that the probability that exactly one of them occurs is $\frac{2}{5}$ and the probability that $A$ or $B$ occurs is $\frac{1}{2}$ then the probability of both of them occur together is

- [JEE MAIN 2020]

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

$A$ and $B$ are two events such that $P(A)=0.54$, $P(B)=0.69$ and $P(A \cap B)=0.35.$ Find $P \left( A \cap B ^{\prime}\right)$ .

If $A$ and $B$ are arbitrary events, then