Question

Suppose you are dealt a hand of 2 cards from a shuffled deck.

Let XX be the number of Kings in your hand. Let YY be the number of Diamonds in your hand.

a. Find the joint probability P( X=2 , Y = 0 )P(X=2,Y=0).

b. Find the joint probability P( X=2 , Y = 2 )P(X=2,Y=2).

Answer 1

a.probability P( X=2 , Y = 0 )

In  hand of 2 cards we want no card from Diamonds y=0 but no. of king should be 2

hence  

P( X=2 , Y = 0 ) = p(C1)*p(C2/C1)

where c1=first card is king and not diamond

c2=second  card is king and not diamond

P( X=2 , Y = 0 ) =p(C1)*p(C2/C1) = (3/39)*((3-1)/(39-1))

= (3/39)*(2/38)

= 6/(39*38)

=0.00405

P( X=2 , Y = 0 ) = (3/39)*(2/38) =0.00405

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b) probability P( X=2 , Y = 2 )

this situations not possible because if we have 2 diamond then we will not have two king because diamond contain only one king

hence P( X=2 , Y = 2 ) =0

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