Q. 295.0( 2 Votes )

A card is drawn a

Answer :

We know that there are 52 cards in a pack of cards.


And there are 13 cards of each suit.


Let p be the probability of drawing a card of heart from a pack of 52 cards.


Then,



[ there are 13 cards of heart in the pack]



Also, p + q = 1


Where if p is the probability of getting a heart out of 52 cards, then q is the probability of not getting a heart out of 52 cards.


q = 1 – p


Putting the value of p in the above equation, we get





Now, let the card be drawn n times.


And let X denote the number of hearts drawn out of a pack of 52 cards.


Then, the binomial distribution is given by,


P (X = r) = nCrprqn-r


Putting and above, we get


…(A)


Where r = 0, 1, 2, 3… n


(i). We need to find the number of times a card can be drawn so that at least there is an even chance of drawing a heart.


In simple words, since n is the number of times a card is drawn, we need to find the smallest n for which P (X = 0) is less than 1/4 satisfies.


So,



From equation A, we have







First, put n = 0.




But 1 0.25


Now, put n = 1.



0.75 < 0.25


But 0.75 0.25


Now, put n = 2.




0.5625 < 0.25


But 0.5626 0.25


Now, put n = 3.




0.42 < 0.25


But 0.42 0.25


Now, put n = 4.




0.31 < 0.25


But 0.31 0.25


Now, put n = 5.




0.23 < 0.25


Thus, smallest n = 5.


, we must draw cards at least 5 times.


(ii). We need to find the number of times a card must be drawn so that the probability of drawing a heart is more than 3/4.


If P (X = 0) is the probability of not drawing a heart at all.


Then, 1 – P (X = 0) is the probability of drawing a heart out of 52 cards.


According to the question,













First, put n = 0.




But 1 0.25


Now, put n = 1.



0.75 < 0.25


But 0.75 0.25


Now, put n = 2.




0.5625 < 0.25


But 0.5626 0.25


Now, put n = 3.




0.42 < 0.25


But 0.42 0.25


Now, put n = 4.




0.31 < 0.25


But 0.31 0.25


Now, put n = 5.




0.23 < 0.25


Thus, smallest n = 5.


, we must draw cards at least 5 times.


Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
Try our Mini CourseMaster Important Topics in 7 DaysLearn from IITians, NITians, Doctors & Academic Experts
Dedicated counsellor for each student
24X7 Doubt Resolution
Daily Report Card
Detailed Performance Evaluation
caricature
view all courses
RELATED QUESTIONS :

A box has 100 penMathematics - Exemplar

State True or FalMathematics - Exemplar

A random variableMathematics - Board Papers

How many times muMathematics - Board Papers

A card from a pacMathematics - Board Papers

An experiment sucMathematics - Board Papers

A bag A contains Mathematics - Board Papers

On a multiple choMathematics - Board Papers

Mark the correct RD Sharma - Volume 2