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? (i) E : the card drawn is a spade F : the card drawn is an ace (ii) E : the card drawn is black F : the card drawn in a king

Answer

N/a

Explanation

(i) There are 13 cards of spades and 4 cards of aces in a pack of 52 cards.
Given that `E=` event of drawing a spade
`impliesn(E)=13`
and `F=` event of drawing an ace
`=n(F)=4` and `n(S)=52`
Here `P(E)=(n(E))/(n(S))=13/52=1/4`
`P(F)=4/52=1/13`
and `EnnF=` event of drawing an ace of spade.
`impliesP(EnnF)=(n(EnnF))/(n(S))=1/52`
Now `P(E)xxP(F)=1/4xx1/13=1/52=P(EnnF)`
Therefore E and F are independent events.
(ii) There are 26 black cards and 4 cards to king in a pack of 52 cards.
Given that `E=` event of drawing a black card
`impliesn(E)=26`
and `F=` event of drawing a king.
`impliesn(F)=4`
and `n(S)=52`
`P(E)=(n(E))/(n(S))=26/52=1/2`
and `P(F)=(n(F))/(n(S))=4/52=1/13`
and `EnnF` event of drawing a black king.
`impliesP(EnnF)=(n(EnnS))/(n(S))=2/52=1/26`
Now, `P(E)xxP(F)=1/2xx1/13=1/26=P(EnnF)`
`impliesP(EnnF)=P(E)P(F)`
Therfore E and F are independent events.
(iii) There are 4 kings, 4 queen and 4 jack in a pack of 52 cards.
`P(E)=P` (card drawn is a king of a queen)
`=P` (king) `+ P` (queen)
`=4/52+4/52=8/52=2/13`
and `P(F)=P` (card drawn is a queen or a jack)
`=P` (queen) `+P` (jack)
`=4/52+4/52=8/52=2/13`
and `EnnF=` event of drawing a queen
`impliesP(EnnF)=4/52=1/13`
Now, `P(E)xxP(F)=2/13xx2/13=4/169!=P(EnnF)`
`impliesP(EnnF)!=P(E)P(F)`
Therefore E and F are not independent events.