In order to understand the probability and sampling distributions of random variables that are the foundation for statistical inference, we will first talk about the basics of probability. Conditional probability and the concept of independence will be introduced.

Hello, everyone. Now that you have finished reading Lesson 3, I would like to provide a brief example to illustrate important points in Lesson three.

There are 8 eggs remaining in a cartoon. 2 are rotten. Randomly pick 1 egg from the 8 eggs, what is the chance that it is rotten?

Since there are 2 rotten eggs out of 8 eggs, the chance of picking an rotten egg is 2/8.

Now, if we randomly pick 2 eggs, what is the chance that both eggs are rotten?

The answer will depend upon after picking the first egg, whether you put it back and thus it has chance of being drawn again.

If you put the first egg back, that is called sample with replacement, then for the second pick, there are still 2 rotten eggs out of 8 eggs. Thus, the probability that both eggs are rotten is 2/8 times 2/8 = 0.0625

If for the second pick, you do not put the first egg back, that is called sample without replacement. Now, when the first pick is rotten, the second pick there is only 1 rotten egg out of 7. Thus the probability that both eggs are rotten is 2/8 times 1/7=0.0357.

The answers from with or without replacement are very different.

We arrive the solution using the same general formula, P(A intersect B) = P(A)*P(B given A).

In the special case when A, B are independent, the multiplication formula becomes simpler. P(A intersects B) = P(A)*P(B). We should keep in mind that this only works when A, B are independent.

When sample with replacement, the first pick and the second pick are independent and P(B given A) = P(B).

When sample without replacement, the first and second pick are dependent (not independent).

Some remarks: A intersect B is more restrictive than A , B. Its probability is a multiplication, getting smaller than or equal to the probability of each event.

A union B is A or B, which includes both A or B. The probability is larger than or equal to the probability of each event.