Answer:

Given:

Total number of balls drawn = 3

Balls are drawn at random with out replacement from a box.

Number of White = 2

Number of Red = 3

Number of Black = 4

x = Number of white balls drawn

y = Number of red balls drawn.

To Find:

The joint probability distribution of (x,y).

Also verify x and y are independent or not.

Solution:

Joint Probability denotes the probability of the occurrence of both the events together.

Total number of balls  = 9.

Total number of balls to be chosen = 3.

• Joint probability of x and y = P(X =x ,Y = y)
• X = { 0 , 1 , 2 }
• Y = { 0 , 1 , 2 , 3 }

P(X = x , Y = y ) = P(0,0) , P(0,1) , P(0,2), P(0,3) , P(1,0) , P(1,1) , P(1,2), P(2,0) ,P(2,1)  .

• P(0,0) = All Black = 4/9  x 3/8 x 2/7 = 24 / 504
• P(0,1) = 2 B , 1 R = 4/9 x 3/8 x 3/7 = 36 / 504
• P(0,2) = 1B, 2R = 4/9 x 3/8 x 2/7 = 24 / 504
• P(0,3) = 3R = 3/9 x 2/8 x 1/7 = 6 / 504

• P(1,0) = 1White , 2 Black = 2/9  x 4/8 x 3/7 = 24 / 504
• P(1,1) = 1W,1R,1B = 2/9 x 3/8 x 4/7 = 24 / 504
• P(1,2) = 1W, 2R = 2/9 x 3/8 x 2/7 =  12/ 504

• P(2,0) = 2 white, 1 Black = 2/9  x 1/8 x 4/7 = 8 / 504
• P(2,1) = 2 W , 1 R = 2/9 x 1/8 x 3/7 = 6 / 504

Checking Dependency,

• P( x ∩ Y ) = 2/9 x 3/8 = 6/72
• P(x ) = 2/9
• P(y) = 3/9
• P(x∩y) ≠ P(x)P(y)
• Therefore x and y are not independent.

Joint probability distribution of (x,y) is found. Also  x and y are not independent.