Question

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 is a king'
(iii)E:'the card drawn is a king or queen'
F:'the card drawn is a queen or jack'

Answer 1

\(S\)={All the 52 cards}⇒\(n(S)=52 \)
(i)\(E\)={13 spades}⇒\(n(E)=13\)
\(∴P(E)=\frac{n(E)}{n(S)}=\frac{13}{52}=\frac{1}{4}\)
\(F\)={4 aces}\(⇒n(F)=4\)
\(∴P(F)=\frac{n(F)}{n(S)}=\frac{4}{52}=\frac{1}{13}\)
Now \(E∩F=\){An ace of spade}\(⇒n(E∩F)=1\)
\(∴P(E∩F)=\frac{n(E∩F)}{n(S)}=\frac{1}{52}\)
Also.\(P(E).P(F)=\frac{1}{4}×\frac{1}{13}=\frac{1}{52}\)
Therefore,\(P(E∩F)=P(E).P(F)\)
Hence,\(E\) and \(F\) are independent events.
(ii)\(E\)={26 black cards}\(⇒n(E)=26\)
\(∴P(E)=\frac{n(E)}{n(S)}=\frac{26}{52}=\frac{1}{2}\)
\(F=\){4 kings}\(⇒n(F)=4\)
\(∴P(F)=\frac{n(F)}{n(S)}=\frac{4}{52}=\frac{1}{13}\)
Now,\(E∩F=\){2 black kings}\(⇒n(E∩F)=2\)
\(∴P(E∩F)=\frac{n(E∩F)}{n(S)}=\frac{2}{52}=\frac{1}{26}\)
Also,\(P(E).P(F)=\frac{1}{2}×\frac{1}{13}=\frac{1}{26}\)
Therefore,\(P(E∩F)=P(E).P(F)\)
Hence,\(E\) and \(F\) are independent events.
(iii)\(E\)={4kings,4queens}\(⇒n(E)=8\)
\(∴P(E)=\frac{n(E)}{n(S)}=\frac{8}{52}=\frac{2}{13}\)
\(F\)={4queens,4jacks}\(⇒n(F)=8\)
\(∴P(F)=\frac{n(F)}{n(S)}=\frac{8}{52}=\frac{2}{13}\)
Now \(E∩F=\){4queens}\(⇒n(E∩f)=4\)
\(∴P(E∩F)=\frac{n(E∩F)}{n(S)}=\frac{4}{52}=\frac{1}{13}\)
Also,\(P(E).P(F)=\frac{2}{13}×\frac{2}{13}=\frac{4}{169}\)
Therefore,\(P(E∩F)≠P(E).P(F)\)
Hence,\(E\) and \(F\) are not independent events.