3.39 from a sack of fruit containing 3 oranges, 2 apples, and 3 bananas, a random sample of 4 pieces of fruit is selected. if x is the number of oranges and y is the number of apples in the sample, find (a) the joint probability distribution of x and y ; (b) p[(x, y ) ∈ a], where a is the region that is given by {(x, y) | x + y ≤ 2}.

Step-by-step explanation:

Let's start defining the random variables.

X : ''The number of oranges in the sample of 4 fruits''

Y : ''The number of apples in the sample of 4 fruits''

X can assume values 0,1,2,3 because they are 3 oranges in the sack of fruit.

Y can assume values 0,1,2 because they are 2 apples in the sack of fruit.

But for the domain of the joint probability distribution it is not a possible value because this will mean that the sample contains 5 fruits.

Also it is an impossible value because given that we selected the 3 bananas we must complete the sample with an orange or either an apple.

We conclude that

There is not another restriction for the domain of

Let's define the combinatorial number.

The joint probability distribution of X and Y is :

From 3 oranges I selected ''x'' oranges.

From 2 apples I selected ''y'' apples.

From 3 bananas I selected ''4-x-y'' bananas that is the total amount of bananas minus the number of oranges and apples.

We multiply this three combinatorial numbers and then we divide by the total forms to choose 4 fruits between a total of fruits.

Now we listed all the values of :

b) P [(X,Y) ∈ A] where A is the region that is given by {(x, y) | x + y ≤ 2} is equal to the sum of the probability for the (X,Y) points that satisfy A.

P [(X,Y) ∈ A] =

P [(X,Y) ∈ A] =

(2,20) means that the 2 is 2 hours and 20 is 20$. so there both 20$ for 2 hours only the same price. like it says. so, the dunk tank and the bounce house are the same price (20$) for the same amount of time (2 hours).

hope i