Question

Pick two cards at random from a well-shuffled deck of 52 cards (pick them simultaneously so they are not the same card). There are 12 cards considered face cards. There are 4 cards with the value 10. Let X be the number of face cards in your hand. Let Y be the number of 10's in your hand. Explain why X and Y are dependent.

Answer 1

X denotes number of face cards

it can take value 0, 1, 2

P[ X = 0 ] = none of the 12 card is chosen in either draw = ( 40/52 )*( 39/51 ) = 1560/2652 = 0.6118

P[ X = 1 ] = one of the card is face card = 2*( 40/52 )*( 12/51 ) = 960/2652 = 0.3619

P[ X = 2 ] = two of the 12 card is chosen in the draw = ( 12/52 )*( 11/51 ) = 132/2652 = 0.0498

Y denotes number of cards numbered 10

it can take value 0, 1, 2

P[ Y = 0 ] = none of the 4 card is chosen in either draw = ( 48/52 )*( 47/51 ) = 2256/2652 = 0.8507

P[ Y = 1 ] = one of the 4 card is chosen in either draw = 2*( 4/52 )*( 48/51 ) = 384/2652 = 0.1448

P[ Y = 2 ] = two of the 4 card is chosen in the draw = ( 4/52 )*( 3/51 ) = 12/2652 = 0.0045

Now, P[ X = 0 and Y = 0 ] =  none of the 16 card is chosen in either draw = ( 36/52 )*( 35/51 ) = 1260/2652 = 0.4751

X and Y are independent if P[ X = 0 and Y = 0 ] = P[ X = 0 ] * P[ Y = 0 ]

P[ X = 0 ] * P[ Y = 0 ] = 0.8507*0.6118 = 0.5204

which is not equal to P[ X = 0 and Y = 0 ] = 0.4751

Hence, the X and Y are not independent

or, X and Y are dependent