31. 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?

31. i. In a deck of $52$ cards,

$13$ are spades and $4$ are aces.

$\therefore P(E)=P$ (the card drawn is spade) $=\frac{13}{52}=\frac{1}{4}$

$\therefore P(F)=P$ (the card drawn is an ace) $=\frac{4}{52}=\frac{1}{13}$

$\therefore P(E\cap F)=P$ (the card drawn is spade and an ace) $=\frac{1}{52}$

$P(E).P(F)=\frac{1}{4}\times \frac{1}{13}=\frac{1}{52}=P(E\cap F)$

$\Rightarrow P(E).P(F)=P(E\cap F)$

$E$ and $F$ are independent.

ii. In a deck of $52$ cards,

$26$ cards are black and $4$ are kings.

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

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

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

$P(E).P(F)=\frac{1}{2}\times \frac{1}{13}=\frac{1}{26}=P(E\cap F)$

$\Rightarrow P(E).P(F)=P(E\cap F)$

Therefore, the event $E$ and $F$ are independent.

iii. In a deck of $52$ cards,

$4$ are king cards

$4$ are queen cards

$4$ are jack cards

$\therefore P(E)=P$ (the card drawn is king or queen) $=\frac{8}{52}=\frac{2}{13}$

$\therefore P(F)=P$ (the card dr

...more

31. i. In a deck of $52$ cards,

$13$ are spades and $4$ are aces.

$\therefore P(E)=P$ (the card drawn is spade) $=\frac{13}{52}=\frac{1}{4}$

$\therefore P(F)=P$ (the card drawn is an ace) $=\frac{4}{52}=\frac{1}{13}$

$\therefore P(E\cap F)=P$ (the card drawn is spade and an ace) $=\frac{1}{52}$

$P(E).P(F)=\frac{1}{4}\times \frac{1}{13}=\frac{1}{52}=P(E\cap F)$

$\Rightarrow P(E).P(F)=P(E\cap F)$

$E$ and $F$ are independent.

ii. In a deck of $52$ cards,

$26$ cards are black and $4$ are kings.

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

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

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

$P(E).P(F)=\frac{1}{2}\times \frac{1}{13}=\frac{1}{26}=P(E\cap F)$

$\Rightarrow P(E).P(F)=P(E\cap F)$

Therefore, the event $E$ and $F$ are independent.

iii. In a deck of $52$ cards,

$4$ are king cards

$4$ are queen cards

$4$ are jack cards

$\therefore P(E)=P$ (the card drawn is king or queen) $=\frac{8}{52}=\frac{2}{13}$

$\therefore P(F)=P$ (the card drawn is a jack or queen) $=\frac{8}{52}=\frac{2}{13}$

 

Therefore, the event  $E$ and $F$ are not independent.

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<p><strong>31. </strong>i. In a deck of <span title="Click to copy mathml"><math><mrow><mn>5</mn><mn>2</mn></mrow></math></span>&nbsp;cards,</p><p><span title="Click to copy mathml"><math><mrow><mn>1</mn><mn>3</mn></mrow></math></span>&nbsp;are spades and&nbsp;<span title="Click to copy mathml"><math><mrow><mn>4</mn></mrow></math></span>&nbsp;are aces.</p><p><span title="Click to copy mathml"><math><mrow><mo>∴</mo><mi>P</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi></mrow></math></span>&nbsp;(the card drawn is spade)&nbsp;<span title="Click to copy mathml"><math><mrow><mo>=</mo><mfrac><mrow><mn>1</mn><mn>3</mn></mrow><mrow><mn>5</mn><mn>2</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></math></span></p><p><span title="Click to copy mathml"><math><mrow><mo>∴</mo><mi>P</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi></mrow></math></span>&nbsp;(the card drawn is an ace)&nbsp;<span title="Click to copy mathml"><math><mrow><mo>=</mo><mfrac><mrow><mn>4</mn></mrow><mrow><mn>5</mn><mn>2</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>1</mn><mn>3</mn></mrow></mfrac></mrow></math></span></p><p><span title="Click to copy mathml"><math><mrow><mo>∴</mo><mi>P</mi><mo stretchy="false">(</mo><mi>E</mi><mo>∩</mo><mi>F</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi></mrow></math></span>&nbsp;(the card drawn is spade and an ace)&nbsp;<span title="Click to copy mathml"><math><mrow><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>5</mn><mn>2</mn></mrow></mfrac></mrow></math></span></p><p><span title="Click to copy mathml"><math><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>.</mo><mi>P</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mo>×</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>1</mn><mn>3</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>5</mn><mn>2</mn></mrow></mfrac><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>E</mi><mo>∩</mo><mi>F</mi><mo stretchy="false">)</mo></mrow></math></span></p><p><span title="Click to copy mathml"><math><mrow><mo>⇒</mo><mi>P</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>.</mo><mi>P</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>E</mi><mo>∩</mo><mi>F</mi><mo stretchy="false">)</mo></mrow></math></span></p><p><br><span title="Click to copy mathml"><math><mrow><mi>E</mi></mrow></math></span>&nbsp;and&nbsp;<span title="Click to copy mathml"><math><mrow><mi>F</mi></mrow></math></span>&nbsp;are independent.</p><p>ii. In a deck of <span title="Click to copy mathml"><math><mrow><mn>5</mn><mn>2</mn></mrow></math></span>&nbsp;cards,</p><p><span title="Click to copy mathml"><math><mrow><mn>2</mn><mn>6</mn></mrow></math></span>&nbsp;cards are black and&nbsp;<span title="Click to copy mathml"><math><mrow><mn>4</mn></mrow></math></span>&nbsp;are kings.</p><p><span title="Click to copy mathml"><math><mrow><mo>∴</mo><mi>P</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi></mrow></math></span>&nbsp;(the card drawn is black)&nbsp;<span title="Click to copy mathml"><math><mrow><mo>=</mo><mfrac><mrow><mn>2</mn><mn>6</mn></mrow><mrow><mn>5</mn><mn>2</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></math></span></p><p><span title="Click to copy mathml"><math><mrow><mo>∴</mo><mi>P</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi></mrow></math></span>&nbsp;(the card drawn is a king)&nbsp;<span title="Click to copy mathml"><math><mrow><mo>=</mo><mfrac><mrow><mn>4</mn></mrow><mrow><mn>5</mn><mn>2</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>1</mn><mn>3</mn></mrow></mfrac></mrow></math></span></p><p><span title="Click to copy mathml"><math><mrow><mo>∴</mo><mi>P</mi><mo stretchy="false">(</mo><mi>E</mi><mo>∩</mo><mi>F</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi></mrow></math></span>&nbsp;(the card drawn is black king)&nbsp;<span title="Click to copy mathml"><math><mrow><mo>=</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mn>5</mn><mn>2</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><mn>6</mn></mrow></mfrac></mrow></math></span></p><p><span title="Click to copy mathml"><math><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>.</mo><mi>P</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>×</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>1</mn><mn>3</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><mn>6</mn></mrow></mfrac><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>E</mi><mo>∩</mo><mi>F</mi><mo stretchy="false">)</mo></mrow></math></span></p><p><span title="Click to copy mathml"><math><mrow><mo>⇒</mo><mi>P</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>.</mo><mi>P</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>E</mi><mo>∩</mo><mi>F</mi><mo stretchy="false">)</mo></mrow></math></span></p><p>Therefore, the event&nbsp;<span title="Click to copy mathml"><math><mrow><mi>E</mi></mrow></math></span>&nbsp;and&nbsp;<span title="Click to copy mathml"><math><mrow><mi>F</mi></mrow></math></span>&nbsp;are independent.</p><p>iii. In a deck of <span title="Click to copy mathml"><math><mrow><mn>5</mn><mn>2</mn></mrow></math></span>&nbsp;cards,</p><p><span title="Click to copy mathml"><math><mrow><mn>4</mn></mrow></math></span>&nbsp;are king cards</p><p><span title="Click to copy mathml"><math><mrow><mn>4</mn></mrow></math></span>&nbsp;are queen cards</p><p><span title="Click to copy mathml"><math><mrow><mn>4</mn></mrow></math></span>&nbsp;are jack cards</p><p><span title="Click to copy mathml"><math><mrow><mo>∴</mo><mi>P</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi></mrow></math></span>&nbsp;(the card drawn is king or queen)&nbsp;<span title="Click to copy mathml"><math><mrow><mo>=</mo><mfrac><mrow><mn>8</mn></mrow><mrow><mn>5</mn><mn>2</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mn>1</mn><mn>3</mn></mrow></mfrac></mrow></math></span></p><p><span title="Click to copy mathml"><math><mrow><mo>∴</mo><mi>P</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi></mrow></math></span>&nbsp;(the card drawn is a jack or queen)&nbsp;<span title="Click to copy mathml"><math><mrow><mo>=</mo><mfrac><mrow><mn>8</mn></mrow><mrow><mn>5</mn><mn>2</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mn>1</mn><mn>3</mn></mrow></mfrac></mrow></math></span></p><div><div><picture><source srcset="https://images.shiksha.com/mediadata/images/articles/1734950459phpwOOTNe_480x360.jpeg" media="(max-width: 500px)"><img src="https://images.shiksha.com/mediadata/images/articles/1734950459phpwOOTNe.jpeg" alt="" width="279" height="157"></picture></div></div><p>&nbsp;</p><p>Therefore, the event&nbsp; <span title="Click to copy mathml"><math><mrow><mi>E</mi></mrow></math></span>&nbsp;and&nbsp;<span title="Click to copy mathml"><math><mrow><mi>F</mi></mrow></math></span>&nbsp;are not independent.</p>

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