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One card is drawn at random from a well shuffled deck of 52 cards. In which of the following cases are the events $\mathrm{E}$ and $\mathrm{F}$ independent?
(i) E: 'the card drawn is a spade'
F: 'the card drawn is an ace'
(ii) $\mathbf{E}$ : 'the card drawn is black'
F : 'the card drawn is a king'
(iii) $\mathbf{E}$ : 'the card drawn is a king or queen'
F: 'the card drawn is a queen or jack'.
(i) E: 'the card drawn is a spade'
F: 'the card drawn is an ace'
(ii) $\mathbf{E}$ : 'the card drawn is black'
F : 'the card drawn is a king'
(iii) $\mathbf{E}$ : 'the card drawn is a king or queen'
F: 'the card drawn is a queen or jack'.
Solution:
2047 Upvotes
Verified Answer
$n(\mathrm{~S})=52$
(i) $\mathrm{E}=\{13$ spades $\}, \mathrm{P}(\mathrm{E})=\frac{13}{52}=\frac{1}{4}$
$$
\mathrm{F}=\{4 \text { aces }\}, P(F)=\frac{4}{52}=\frac{1}{13}
$$
$\mathrm{E} \cap \mathrm{F}=\{$ an ace of spade $\}, \mathrm{P}(\mathrm{E} \cap \mathrm{F})=\frac{1}{52}$
Also, $\mathrm{P}(\mathrm{E}), \mathrm{P}(\mathrm{F})=\frac{1}{4} \times \frac{1}{13}=\frac{1}{52}$
$$
\therefore \quad \mathrm{P}(\mathrm{E} \cap \mathrm{F})=\mathrm{P}(\mathrm{E}) \cdot \mathrm{P}(\mathrm{F})
$$
Hence, $\mathrm{E}$ and $\mathrm{F}$ are independent events.
(ii) $\mathrm{E}=\{26$ black cards $\}, \mathrm{P}(\mathrm{E})=\frac{26}{52}=\frac{1}{2}$.
$$
\mathrm{F}=\{4 \text { kings }\}, \mathrm{P}(\mathrm{F})=\frac{4}{52}=\frac{1}{13}
$$
$\mathrm{E} \cap \mathrm{F}=\{2$ black kings $\}$,
$\mathrm{P}(\mathrm{E} \cap \mathrm{F})=\frac{2}{52}=\frac{1}{26}$
Now, $\mathrm{P}(\mathrm{E}) \cdot \mathrm{P}(\mathrm{F})=\frac{1}{2} \times \frac{1}{13}=\frac{1}{26}$ $\therefore \quad \mathrm{P}(\mathrm{E} \cap \mathrm{F})=\mathrm{P}(\mathrm{E}) \cdot \mathrm{P}(\mathrm{F})$.
Hence $\mathrm{E}$ and $\mathrm{F}$ are independent events.
(iii) $\mathrm{E}=\{4$ kings, 4 queens $\}, n(\mathrm{E})=8$
$$
\mathrm{P}(\mathrm{E})=\frac{n(E)}{n(S)}=\frac{8}{52}=\frac{2}{13}
$$
$\mathrm{F}=\{4$ queens, 4 jacks $\}, \mathrm{P}(\mathrm{F})=\frac{8}{52}=\frac{2}{13}$
$\mathrm{E} \cap \mathrm{F}=\{4$ queens $\}$
$\mathrm{P}(\mathrm{E} \cap \mathrm{F})=\frac{4}{52}=\frac{1}{13}$
Now, $\mathrm{P}(\mathrm{E}) \cdot \mathrm{P}(\mathrm{F})=\frac{2}{13} \times \frac{2}{13}=\frac{4}{169}$
Thus $\mathrm{P}(\mathrm{E} \cap \mathrm{F}) \neq \mathrm{P}(\mathrm{E}) . \mathrm{P}(\mathrm{F})$
Hence $\mathrm{E}$ and $\mathrm{F}$ are not independent events.
(i) $\mathrm{E}=\{13$ spades $\}, \mathrm{P}(\mathrm{E})=\frac{13}{52}=\frac{1}{4}$
$$
\mathrm{F}=\{4 \text { aces }\}, P(F)=\frac{4}{52}=\frac{1}{13}
$$
$\mathrm{E} \cap \mathrm{F}=\{$ an ace of spade $\}, \mathrm{P}(\mathrm{E} \cap \mathrm{F})=\frac{1}{52}$
Also, $\mathrm{P}(\mathrm{E}), \mathrm{P}(\mathrm{F})=\frac{1}{4} \times \frac{1}{13}=\frac{1}{52}$
$$
\therefore \quad \mathrm{P}(\mathrm{E} \cap \mathrm{F})=\mathrm{P}(\mathrm{E}) \cdot \mathrm{P}(\mathrm{F})
$$
Hence, $\mathrm{E}$ and $\mathrm{F}$ are independent events.
(ii) $\mathrm{E}=\{26$ black cards $\}, \mathrm{P}(\mathrm{E})=\frac{26}{52}=\frac{1}{2}$.
$$
\mathrm{F}=\{4 \text { kings }\}, \mathrm{P}(\mathrm{F})=\frac{4}{52}=\frac{1}{13}
$$
$\mathrm{E} \cap \mathrm{F}=\{2$ black kings $\}$,
$\mathrm{P}(\mathrm{E} \cap \mathrm{F})=\frac{2}{52}=\frac{1}{26}$
Now, $\mathrm{P}(\mathrm{E}) \cdot \mathrm{P}(\mathrm{F})=\frac{1}{2} \times \frac{1}{13}=\frac{1}{26}$ $\therefore \quad \mathrm{P}(\mathrm{E} \cap \mathrm{F})=\mathrm{P}(\mathrm{E}) \cdot \mathrm{P}(\mathrm{F})$.
Hence $\mathrm{E}$ and $\mathrm{F}$ are independent events.
(iii) $\mathrm{E}=\{4$ kings, 4 queens $\}, n(\mathrm{E})=8$
$$
\mathrm{P}(\mathrm{E})=\frac{n(E)}{n(S)}=\frac{8}{52}=\frac{2}{13}
$$
$\mathrm{F}=\{4$ queens, 4 jacks $\}, \mathrm{P}(\mathrm{F})=\frac{8}{52}=\frac{2}{13}$
$\mathrm{E} \cap \mathrm{F}=\{4$ queens $\}$
$\mathrm{P}(\mathrm{E} \cap \mathrm{F})=\frac{4}{52}=\frac{1}{13}$
Now, $\mathrm{P}(\mathrm{E}) \cdot \mathrm{P}(\mathrm{F})=\frac{2}{13} \times \frac{2}{13}=\frac{4}{169}$
Thus $\mathrm{P}(\mathrm{E} \cap \mathrm{F}) \neq \mathrm{P}(\mathrm{E}) . \mathrm{P}(\mathrm{F})$
Hence $\mathrm{E}$ and $\mathrm{F}$ are not independent events.
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