Question

1) One card is drawn at random from a well shuffled deck of $52$ cards. In which of the following cases are the events
$E\text{and}F$ independent?
$i)$ $E:\text{'the card drawn is a spade'}$
$F:{\text{'the card drawn is an ace}}^{\prime }$

2) One card is drawn at random from a well shuffled deck of $52$ cards. In which of the following cases are the events $E\text{and}F$ independent?
$ii)$ $E:\text{'the card drawn is black'}$
$F:{\text{'the card drawn is a king}}^{\prime }$

3) One card is drawn at random from a well shuffled deck of $52$ cards. In which of the following cases are the events $EandF$ independent?
$iii)$ $E:\text{'the card drawn is a king or queen'}$
$F:\text{'the card drawn is a queen or jack'}$

Answer 1

1) In a deck of $52$ cards,

$13$ cards are spades and $4$ cards are aces.

$E:\text{'The card drawn is a spade'}$

$P\left(E\right)=\frac{13}{52}$

$F:\text{'The card drawn is an ace'}$

$P\left(F\right)=\frac{4}{52}$

Also
$P(E\cap F)=P\left(\text{The card drawn is spade and an ace}\right)$

$P(E\cap F)=\frac{1}{52}$

We know that,
For independent events,

$P(E\cap F)=P\left(E\right)\times P\left(F\right)....\left(1\right)$

$\text{Substituting the value of}P\left(E\right)\&P\left(F\right)\text{in}\left(1\right),$

$\frac{1}{52}=\frac{13}{52}\times \frac{4}{52}$

$\frac{1}{52}=\frac{1}{52}$

$\therefore L.H.S.=R.H.S.$

Therefore, $E\&F$ are independent events.

2) In a deck of $52$ cards,

$26$ cards are black, and 4 cards are king.

$E:$ 'The card drawn is black'

$P\left(E\right)=\frac{26}{52}$

$F:$ 'The card drawn is a king'

$P\left(F\right)=\frac{4}{52}$

Also,
$P(E\cap F)=P\left(\text{The card drawn is a black king}\right)$

$P(E\cap F)=\frac{2}{52}$

We know that,

For independent events,

$P(E\cap F)=P\left(E\right)\times P\left(F\right)...\left(1\right)$

Substituting the value of $P\left(E\right)\&P\left(F\right)\text{in}\left(1\right),$

$\frac{2}{52}=\frac{26}{52}\times \frac{4}{52}$

$\frac{1}{26}=\frac{1}{26}$

$\therefore L.H.S=R.H.S$

Therefore, $E\&F$ are independent events.

3) In a deck a $52$ cards,

$4$ cards are kings, $4$ cards are queens, and $4$ cards are jacks.

From Question,

$E:$ 'The card drawn is a king or queen'

$P\left(E\right)=\frac{8}{52}$

$F:$ 'The card drawn is a queen or jack'

$P\left(F\right)=\frac{8}{52}$

Also,

$P(E\cap F)=P\left(\text{card drawn is a queen}\right)$

$P(E\cap F)=\frac{4}{52}$

We know that, For independent events

$P(E\cap F)=P\left(E\right)\times P\left(F\right)...\left(1\right)$

Now, $P\left(E\right)\times P\left(F\right)=\frac{8}{52}\times \frac{8}{52}$

$P\left(E\right)\times P\left(F\right)=\frac{4}{169}$

$\therefore P(E\cap F)\ne P\left(E\right)\times P\left(F\right)$

Therefore, $E\&F$ are not independent events.