Number divisible by 3 & 13 + two expected value -questions

[Unelmoija]

Hi!


I'm studying for entrance exams. Hopefully in a couple of years I'll be a licensed psychologist! I'll be posting questions from time to time. I'm thrilled that a site like this exists, because I'm sure there's a lot of demand for math help.

Thanks for your answers in advance



Divisible by....


You randomly select one integer out of the first one hundred integers. What probability there is that you can divide that number with 3 and 13?


Comments:

Is there an easy way to calculate this? I wrote down the first one hundred integers and then selected the ones that were divisible by 3 and 13. Naturally the first number is 3 * 13 = 39, but how do I calculate the second one, which is 52?



Expected value question #1

A target has three zones for points. The radius of the innermost zone is 10 centimeters, the second 20 centimeters and the outermost is 30 centimeters. The inner zone is worth 5 points, the second 3 points and the outermost 1 points.

A dart hits a randomly selected spot on the target board, on which every spot is equally probable. What is the expected value of the points?


Comments:

I'm totally lost with this. The expected value is 2.1. The likelihood of getting 5 points is 1/9, 3 points 3/9 and 1 points 5/9. Is the divisor calculated by 5+3+1 = 9?



Expected value question #2

A shop owner has taken four copies of a weekly magazine to be sold in his store. He has also written down the number of copies sold per week and the probability of the purchases

Number of copies sold	Probability
0	0.10
1	0.20
2	0.30
3	0.30
4	0.10

Every copy costs 3 dollars for the shop owner. He sells the magazines for 5 dollars. Magazines that are left unsold cannot be returned in exchange for money.

How many copies should the shop owner buy to be sold in his store? What is the expected value of his profits?


Comments:

Am I supposed to calculate E(X) = 0 * 0.1 + 1 * 0.2 + 2 * 0.3 + 3 * 0.3 + 4 * 0.1 = 2.1 ? Does this equate to taking two copies of the magazine to be sold? I don't really understand what is the expected value of the profits. The exercise book says it's:

0 copies = 0 $
1 copy = 1.5$
2 copies = 2 $
3 copies = 1 $
4 copies = –1.5 $

 

[lookagain]

Divisible by....You randomly select one integer out of the first one hundred [positive] integers.
What probability there is that you can divide that number with 3 and 13?
Comments:Is there an easy way to calculate this? I wrote down the first one hundred [positive] integers
and then selected the ones that were divisible by 3 and 13. Naturally the first number is 3 * 13 = 39,
but how do I calculate the second one, which is 52?

52 is not divisible by 3.


Expected value question #1

A target has three zones for points. The radius of the innermost zone is 10 centimeters,
the second 20 centimeters and the outermost is 30 centimeters.

The inner zone is worth 5 points, the second 3 points and the outermost 1 points
A dart hits a randomly selected spot on the target board, on which every spot is equally probable.

What is the expected value of the points?Comments:I'm totally lost with this. The expected value is 2.1.

The likelihood of getting 5 points is 1/9, 3 points 3/9 and 1 points 5/9.

Is the divisor calculated by 5+3+1 = 9?

5(1/9) + 3(3/9) + 1(5/9) =

5/9 + 9/9 + 5/9 =

19/9 ~

2.1

First question:

The next multiple of 3, after 3, is 6.

The next multiple of 13, after 13, is 26.6*13 = 78.

And 3*26 = 78.


Either way, the next number is 78.

 

[Unelmoija]

Expected value question #2

A shop owner has taken four copies of a weekly magazine to be sold in his store. He has also written down the number of copies sold per week and the probability of the purchases

Number of copies sold	Probability
0	0.10
1	0.20
2	0.30
3	0.30
4	0.10

Every copy costs 3 dollars for the shop owner. He sells the magazines for 5 dollars. Magazines that are left unsold cannot be returned in exchange for money.

How many copies should the shop owner buy to be sold in his store? What is the expected value of his profits?

I pondered about this problem and maybe I've found a solution. Since the exercise asks the specific number of copies, I made a chart of what is the mixed probability of either selling or not selling the specific number of copies:

0 copies * (0.10 * 0.90) + 1 copy * (0.20 * 0.80) + 2 copies * (0.30 * 0.7) + 3 copies * (0.3 * 0.7) + 4 copies * (0.3 * 0.7) = 1.57, which when rounded is two copies just like the answer is in the exercise book.

Unfortunately I haven't figured out yet how do I calculate the expected value of the profits.