Question

From a pack of $52cards$, face cards and tens are removed and kept aside, then a card is drawn at random from the remaining cards. If A: The events that the card is drawn is an ace. H: The events that the card is drawn is a heart.S: the events that the card is drawn is a spade. Then, which of the following holds?

• A. $9P\left(A\right)=4P\left(H\right)$
• B. $P\left(S\right)=4P(A\cap H)$
• C. $3P\left(H\right)=P(A\cap S)$
• D. $Noneofthese$

Answer 1

Answer: (A)

$9P\left(A\right)=4P\left(H\right)$

$Givenfromapackof52cards,facecardsandtensareremoved$

$\Rightarrow Totalno.ofremainingcards=52-(12+4)=36cards$

$A:Theeventthatthecarddrawnisanace,P\left(A\right)=\frac{4}{36}=\frac{1}{9}$

$H:TheeventthatthecarddrawnisanHeart,P\left(H\right)=\frac{9}{36}=\frac{1}{4}$

$S:TheeventthatthecarddrawnisanSpade,P\left(S\right)=\frac{9}{36}=\frac{1}{4}$

$andP(A\cap S)=\frac{1}{36}$

$P(A\cup S)=P\left(A\right)+P\left(S\right)-P(A\cap S)$

$=\frac{1}{9}+\frac{1}{4}-\frac{1}{36}$

$=\frac{1}{3}$

$P\left(A\right)=\frac{1}{9},P\left(H\right)=\frac{1}{4}$

$\Rightarrow 9P\left(A\right)=1,4P\left(H\right)=1$

$\Rightarrow 9P\left(A\right)=4P\left(H\right)$