Lesson 4 Chapter 2
To tackle and solve the probability problem, there is always a need to count how many elements available in the event and sample space. Here, we discuss some important counting principles and techniques.
Get the source code for this lesson
Counting all possible outcomes
Let's consider the special case of having two experiments as and . The basic principle states that if one experiment () results in N possible outcomes and if another experiment () leads to M possible outcomes, then conducting the two experiments will have possible outcome, in total. Assume experiment has M possible outcomes as and has N possible outcomes as .
It is easy to prove such a principle for its special case. All you need in to count all possible outcomes of two experiments:
The generalized principle of counting can be expressed as below:
Generalized Basic Principle of Counting
What is a permutation? Suppose we have three persons called Michael, Bob, and Alice. Assume the three of them stay in a queue. How many possible arrangements we have? Take a look at the arrangements as follows:
As above, you will see six permutations. Right? But, we cannot always write all possible situations! We need some math. The intuition behind this problem is that we have three places to fill in a queue when we have three persons. For the first place, we have three choices. For the second place, there are two remaining choices. Finally, there is only one choice left for last place! So we can extend this conclusion to the experiment that we have choices. Hence, we get the following number of permutations:
which you can compute by Python as below:
The combination stands for different combinations of objects from a larger set of objects. For example, assume we have a total number of objects. With how many ways can we select objects from that objects? Let's get back to the above examples. Assume we have three candidates named Michael, Bob, and Alice, and we only desire to select two candidates. How many different combinations of candidates exist?
Let's get back to the general question: How many selections we can have if we desire to pick objects from objects?
The above definition can be generalized.