Solution:
Given: A deck of 52 cards.
Concept: Two events are independent, statistically independent, or stochastically independent if the occurrence of one does not affect the probability of occurrence of the other.
Calculation:
(i) In a deck of 52 cards, there are 13 spades, 4 aces, and only one card that is both a spade and an ace.
Hence, $P(E)=$ the card drawn is a spade $=13 / 52=1 / 4$
$\mathrm{P}(\mathrm{F})=$ the card drawn is an ace $=4 / 52=1 / 13$
$P(E \cap F)=$ the card drawn is a spade and ace both $=1 / 52 \ldots . .$ (1)
And $P(E) . P(F)$
$=1 / 4 \times 1 / 13=1 / 52 \ldots .(2)$
From (1) and (2)
$\Rightarrow P(E \cap F)=P(E) . P(F)$
Hence, $\mathrm{E}$ and $\mathrm{F}$ are independent events.
(ii) In a deck of 52 cards, 26 cards are black, 4 cards are king, and there are only two cards that are both black and king at the same time.
Hence, $P(E)=$ the card drawn is of black $=26 / 52=1 / 2$
$P(F)=$ the card drawn is a king $=4 / 52=1 / 13$
$P(E \cap F)=$ the card drawn is a black and king both $=2 / 52=1 / 26 \ldots$ (1)
And $P(E) . P(F)$
$=1 / 2 \times 1 / 13=1 / 26 \ldots .(2)$
From (1) and (2)
$\Rightarrow P(E \cap F)=P(E) \cdot P(F)$
Hence, $\mathrm{E}$ and $\mathrm{F}$ are independent events.
(iii) In a standard 52-card deck, the queen, the king, and the jack are represented by four cards each.
Hence, $\mathrm{P}(\mathrm{E})=$ the card drawn is either king or queen $=8 / 52=2 / 13$
$\mathrm{P}(\mathrm{F})=$ the card drawn is either queen or jack $=8 / 52=2 / 13$
There are 4 cards which are either king or queen and either queen or jack.
$P(E \cap F)=$ the card draw $n$ is either king or queen and either queen or jack $=4 / 52=1 / 13$ $\ldots$ (1)
And $P(E) . P(F)$
$=2 / 13 \times 2 / 13=4 / 169 . \ldots .$
From (1) and (2)
$\Rightarrow P(E \cap F) \neq P(E) . P(F)$
Hence, $\mathrm{E}$ and $\mathrm{F}$ are not independent events.