Question

From a sack of fruit containing ​3 bananas, 3 oranges​, and 3 apples ​, a random sample of 4 pieces of fruit is selected. Suppose X is the number of bananas and Y is the number of apples in the sample.

​(a) Find the joint probability distribution of X and Y.

​(b) Find ​P[(X,Y)​A], where A is the region that is given by (x,y)

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Part 1

​(a) Complete the joint probability distribution below.

​(Type integers or simplified​ fractions.)

Answer 1

Solution:

(a):

$P(X=x,Y=y)=\dfrac{^3C_x\ ^3C_y\ ^3C_{4-x-y}}{^9C_4}; x=0,1,2,3,y=0,1,2,3$

So, $P(X=0,y=1)=\dfrac{^3C_0\ ^3C_1\ ^3C_{4-0-1}}{^9C_4}=0.0238$

$P(X=1,y=1)=\dfrac{^3C_1\ ^3C_1\ ^3C_{4-1-1}}{^9C_4}=0.2143$

and so on till $P(X=4,y=4)=\dfrac{^3C_4\ ^3C_4\ ^3C_{4-4-4}}{^9C_4}=0$

(b):

$P[(x,y),x+y\le1]=f(0,0)+f(0,1)+f(1,0) \\=0+0.0238+0.0238 \\=0.0476$