14.Probability
easy

સારી રીતે ચીપેલાં $52$ પત્તાંની થોકડીમાંથી એક પનું યાદચ્છિક રીતે પસંદ કરવામાં આવે છે. ઘટનાઓ $E$ અને $F$ નિરપેક્ષ છે ?

$E :$ ‘પસંદ કરેલ પતું કાળીનું છે'. $F :$ ‘પસંદ કરેલ પતું એક્કો છે'. 

Option A
Option B
Option C
Option D

Solution

In a deck of $52$ cards, $13$ cards are spades and $4$ cards are aces.

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

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

In the deck of cards, only $1$ card is an ace of spades.

$ \mathrm{P}(\mathrm{EF})=\mathrm{P}$ (the card drawn is spade and an ace) $=\frac {1}{52}$

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

$\Rightarrow \mathrm{P}(\mathrm{E}) \times \mathrm{P}(\mathrm{F})=\mathrm{P}(\mathrm{EF})$

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

Standard 11
Mathematics

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