There are 10 balls in an urn, numbered from 1 to 10. if 5 balls are selected at random and their numbers are added, what is the expected value of the total?

Let denote the value on the -th drawn ball. We want to find the expectation of , which by linearity of expectation is

(which is true regardless of whether the are independent!)

At any point, the value on any drawn ball is uniformly distributed between the integers from 1 to 10, so that each value has a 1/10 probability of getting drawn, i.e.

and so

Then the expected value of the total is

The correct are is 5.5 cm².

Step-by-step explanation:

First of all, Carlos calciulated the are of the rectangle correctly, only the are unit is wrong, it should be 3 cm².

He made a mistake calculating the area of a the triangle: The area of a triangle can be calculated by multiplying the base by the height by .

(Comment if you want a deeper explanation on where this comes from)

He mistakenly takes 4 cm for the height, but if you look carefully the height of the triangle in this case is 4 cm - 1.5 cm = 2.5 cm.

That gives for the area of the triangle.

Adding those two together gets you a total area of 5.5 cm².