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

The variables X and Y are dependent.

Explanation:

The variable X denotes number of face cards . That is it can take values,

X = {0, 1, 2 }

Compute the probability for all he values of X as follows:

P [X = 0] = P (None of the 12 card is chosen in either draw)

= (40/52)×(39/51)

= 1560/2652

= 0.5882

P [X = 1] = P (One of the card is face card is selected)

= 2×(40/52)×(12/51)

= 960/2652

= 0.3619

P [X = 2] = P (Two of the 12 card is chosen in the draw)

= (12/52)×(11/51)

= 132/2652

= 0.0498

The variable Y denotes number of cards numbered 10 . Thus, it can take values:

Y = {0, 1, 2 }

P [Y = 0] = P (None of the 4 card is chosen in either draw)

= (48/52)×(47/51)

= 2256/2652

= 0.8507

P [Y = 1] = P (One of the 4 card is chosen in either draw)

= 2×(4/52)×(48/51)

= 384/2652

= 0.1448

P [Y = 2] = P (Two of the 4 card is chosen in the draw)

= (4/52)×(3/51)

= 12/2652

= 0.0045

Now compute the probability of (X and Y).

P [X = 0 and Y = 0] = P(None of the 16 card is chosen in either draw)

= (36/52)×(35/51)

= 1260/2652

= 0.4751

The variables 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 × 0.5882

= 0.5204

The two values are not equal.

Hence, the variables X and Y are not independent.