Homework 2

	Student A
Full Name	Ido Fang Bentov
ID	CLASSIFIED
Email	CLASSIFIED

---

---

Question 1

The probability that a part produced in chamber is defective is and the probability that a part produced in chamber is defective is . All parts are independent of each other. If we know that exactly two of the parts produced in one shift are defective, what is the probability that in a random sample of five parts selected from these parts, exactly two are defective?

Solution:
The number of ways of selecting parts from is:

Knowing we already picked two defective parts, we now have parts to select from, and more to pick. The number ways of picking them are:

Therefore, the probability is:

Question 2

Concrete samples from a building site are sent to one of four laboratories (, , , ), selected at random. When a sample arrive at the lab, three properties are tested (,,).

On a particular day, two samples are sent. The following events are defined:

• : both samples are sent to
• : at least one sample is sent to
• : both samples are sent to the same lab

Part a

Find the probabilities .

Solution:
We define:

• sample is sent to

Since the the labs are selected at random, there are labs, and the selections are independent:

is a subset of , so:

To calculate we can use the additive rule:

The intersection of and corresponds to the event that both samples are sent to , that is, event . Hence, , so we can write:

Part b

If the result of the test of does not meet the specification, we say that failed. When a sample is tested in ,:

For a single sample tested in , find the probabilities

Solution:
We’ll use the shorthand:

• means .

Using the additive rule:

Applying the same logic, we get:

For the last probability, we get:

Since , we know that . Therefore: