logo

Ask Questions, Get Answers

 
X
 Search
Want to ask us a question? Click here
Browse Questions
Ad
Home  >>  CBSE XII  >>  Math  >>  Probability
0 votes

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.

Can you answer this question?
 
 

1 Answer

0 votes
Toolbox:
  • If A and B are independent events, \(P(A\cap\;B)=P(A)\;P(B)\)
(i) E : the card drawn is black F : the card drawn is a king
In a deck there are 52 cards - 26 of each color and and there are 4 kings, one in each suit, and 2 of each color.
Given E: the card drawn is black, P(E) = $\large\frac{26}{52} = \frac{1}{2}$
Given F: the card drawn is a king, P(F) = $\large\frac{4}{52} = \frac{1}{13}$
If A and B are independent events, \(P(A\cap\;B)=P(A)\;P(B)\)
P (black card $\cap$ king ) = P (E $\cap$ F) = P (drawing an black king) = $\large\frac{2}{52} = \frac{1}{26}$
Now, P (E $\cap$ F) = P(E) $\times$ P(F) = $\large\frac{1}{2}$$\;\times$$\large\frac{1}{13} = \frac{1}{26}$
Therefore E and F are independent
(ii) E : the card drawn is a spade F : the card drawn is an ace
In a deck there are 52 cards - 13 of each suit - spades, clubs, diamonds and hearts, and there are 4 aces, one in each suit.
Given E: the card drawn is a spade, P(E) = $\large\frac{13}{52} = \frac{1}{4}$
Given F: the card drwan is an ace, P(F) = $\large\frac{4}{52} = \frac{1}{13}$
P (spade $\cap$ ace) = P (E $\cap$ F) = P (drawing an ace of spades) = $\large\frac{1}{52}$
P (E $\cap$ F) = P(E) $\times$ P(F) = $\large\frac{1}{4}$$\;\times$$\large\frac{1}{13} = \frac{1}{52}$
Therefore E and F are independent
(iii) E : the card drawn is a king or queen F : the card drawn is a queen or jack
Given a deck of 52 cards, there are 4 kings, 4 queens and 4 jacks, and the P(drawing a king) = P(drawin a queen) = P(drawing a jack) = $\large\frac{4}{52} = \frac{1}{13}$
Given E: the card drawn is a king or queen, P (E) = P(king) + P(queen) = $\large\frac{1}{13} + \frac{1}{13} = \frac{2}{13}$
Given F: the card drawn is a queen or a jack, P(F) = P(queen) + P(jack) = = $\large\frac{1}{13} + \frac{1}{13} = \frac{2}{13}$
P (E $\cap F) = P (card drawn is a queen) = $\large\frac{4}{52} = \frac{1}{13}$
P (E $\cap$ F) = P(E) $\times$ P(F) = $\large\frac{2}{13}$$\;\times$$\large\frac{2}{13} = \frac{4}{169} $$\;\neq \large\frac{1}{13}$
Therefore E and F are NOT independent
answered Jun 19, 2013 by balaji.thirumalai
 

Related questions

Ask Question
student study plans
x
JEE MAIN, CBSE, NEET Mobile and Tablet App
The ultimate mobile app to help you crack your examinations
Get the Android App
...