Counting Principles in Probability | CFA Level I Quantitative Methods
Welcome back! In this lesson, we will explore the principles of counting and how they relate to probability. Understanding counting principles helps us determine the total number of possibilities in a problem, which is crucial for probability calculations.
Multiplication Rule of Counting
Let’s start with the multiplication rule of counting. Suppose we have 3 steps in an investment decision process:
- Step 1 can be done in 2 ways
- Step 2 can be done in 3 ways
- Step 3 can be done in 2 ways
Using the multiplication rule, we can calculate the total number of ways to carry out these 3 steps: 2 × 3 × 2 = 12 ways.
Factorials and Assignments
Now let’s consider an assignment problem. You have 3 staff members and must assign them to 3 different roles. How many ways can you assign them?
Breaking it down into a 3-step assignment process:
- First step: 3 ways to assign the first staff member
- Second step: 2 ways to assign the second staff member
- Third step: 1 way to assign the remaining staff member
Total number of ways: 3 × 2 × 1 = 6 ways.
This calculation, 3 × 2 × 1, is also known as 3 factorial (3!). In general, n factorial (n!) is the product of all positive integers from n down to 1.
You are an analyst covering 9 stocks. You must label 4 stocks as “Buy,” 3 stocks as “Sell,” and 2 stocks as “Hold.” How many ways can these 9 stocks be labeled, considering that the sequence of labeling does not matter?
The total number of sequences is 9 factorial (9!). Using a calculator, we find there are 362,880 different ways to sequence the stocks. However, THIS IS WRONG! The sequence does not matter, so we must account for this.
To eliminate redundant sequences, we divide the total number of sequences by the product of the redundant sequences: 4! (for Buy), 3! (for Sell), and 2! (for Hold). The correct answer is 1260 ways to label the stocks.
Multinomial and Combination Formulas
The example above demonstrates the multinomial formula. Given n objects to be labeled with k different labels, the number of ways to label the objects is given by:
n! / (n1! × n2! × … × nk!)
A special case of the multinomial formula is the combination formula (n choose r), used when there are only two different labels:
n! / (r! × (n – r)!)
Permutations and Combinations
When the sequence of selection is not important, we use the combination formula. For example, if you have 4 stocks and must select 3 to place in your portfolio, you can calculate the number of combinations with the 4 choose 3 formula, giving you 4 combinations to choose from.
If the sequence is important, you should use the permutation formula:
n! / (n – r)!
In our example, if the sequence of buying the stocks is important, there are 24 permutations to choose 3 stocks from a group of 4.
You can also use the nCr (combinations) and nPr (permutations) functions on your calculator to quickly find your answers.
That wraps up our lesson on counting principles in probability. Remember to practice with quizzes and problems to master this topic. Good luck with your CFA exam preparations, and see you in the next lesson!
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