Answer:
The variables X and Y are dependent.
Step-by-step explanation:
The variable <em>X</em> denotes number of face cards
. That is it can take values,
<em>X</em> = {0, 1, 2
}
Compute the probability for all he values of <em>X</em> 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 <em>Y</em> denotes number of cards numbered 10
. Thus, it can take values:
<em>Y</em> = {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 (<em>X</em> and <em>Y</em>).
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 <em>X</em> and <em>Y</em> 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.
