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] =

step-by-step explanation:

equation

you are asked to write the equation

  expression = 0

using the given expression. when you do that, you have

  2x^2-2x-12 = 0

factoring

you can see that each of the coefficients is even, so a factor of 2 can be removed immediately.

  2(x^2 -x -6) = 0

if you multiply the binomials (x+a)(x+b), you get

  (x +a)(x +b) = x^2 +(a+b)x +ab

this tells you the coefficient of x is the sum of two factors of the constant term. here, that means -1 is the sum of two factors of -6. we want to know those factors so we can factor the trinomial.

  -6 = -1×6 = -2×3 = -3×2 = -6×1

the factor pair that sums to -1 is {-3, 2}, so our factorization is

  2(x -3)(x +2) = 0

solutions

the solutions to this equation are the values of x that make the product zero. a product is zero only when only of the factors of the product is zero. here, that means

  x - 3 = 0

  x = 3 . . . add 3 to both sides of the equation

or

  x +2 = 0

  x = -2 . . . add -2 to both sides of the equation

the solutions to the equation are x=3 or x=-2. both of these values of x satisfy the equation—they make it be true.

8 apples are left.

step-by-step explanation:

10-2=8

10 apples

take 2

now u have 8