Introduction: Why study "Counting and Probability"?

Hello! Let’s dive into one of the biggest milestones in Math A: "Counting and Probability."
You might think, "Math is just boring calculations..." but this topic is incredibly practical. Whether it's the odds of winning a lottery, the chances of pulling a rare character in a smartphone game, or choosing snacks, our daily lives are filled with "what if" possibilities.
It might feel like a puzzle at first, but once you grasp the techniques, it can become a strong subject that helps you score points. Let’s take it one step at a time and have some fun with it!

1. The Basics of Counting: The Addition and Multiplication Rules

First, let’s master the two fundamental rules. If you get these wrong, all your subsequent calculations will be off.

① The Addition Rule (for "or")

When two events A and B cannot happen at the same time, you use addition to find the number of ways A or B can occur.
Example: Rolling a die once—getting a 1 (1 way) or a 6 (1 way). Total: \(1 + 1 = 2\) ways.

② The Multiplication Rule (for "consecutive" events)

When event A occurs, and for each of those cases, event B occurs, you use multiplication to find the number of ways A and B happen in sequence.
Example: If you have 3 types of T-shirts and 2 types of pants, the number of outfit combinations is \(3 \times 2 = 6\) ways.

【Pro-tip!】
When in doubt, remember: "If it's simultaneous/consecutive, multiply (×). If it's separate cases, add (+)."

2. Arrangements: Permutations

The method of choosing \(r\) items from \(n\) distinct items and arranging them in a specific order is called a permutation, denoted by the symbol \(P\).

Permutation Formula

\( nPr = n \times (n-1) \times \dots \times (n-r+1) \)
Example: Choosing 3 people out of 5 to decide the running order for a relay → \( 5P3 = 5 \times 4 \times 3 = 60 \) ways.

About Factorials (!)

When you want to "arrange everyone," you use factorials (!).
\( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)

Special Permutations

  • Circular Permutations: When sitting at a round table, arrangements that are the same when rotated are counted as one. The formula is \( (n-1)! \).
    (The trick is to fix one person in place!)
  • Permutations with Identical Items: When there are identical items mixed in, you divide by the number of duplicates.
    Formula: \( \frac{n!}{p!q!r!} \)

【Common Mistake】
It is important to determine from the problem statement whether you need to "arrange" things or just "select" them. Ask yourself, "Does the order matter?"

3. Selecting: Combinations

The method of choosing \(r\) items from \(n\) distinct items without caring about the order is called a combination, denoted by the symbol \(C\).

Combination Formula

\( nCr = \frac{nPr}{r!} \)
Example: Choosing 2 people for cleaning duty out of 5 (order doesn't matter) → \( 5C2 = \frac{5 \times 4}{2 \times 1} = 10 \) ways.

【Nugget of Wisdom】
Calculating \( 10C8 \) is a hassle, right? Actually, there is a property: \( nCr = nC(n-r) \). In other words, "choosing 8 people out of 10" is the same as "deciding the 2 people who are NOT chosen!"
\( 10C8 = 10C2 = \frac{10 \times 9}{2 \times 1} = 45 \), which is much easier to calculate.

4. Basics of Probability

Probability is found by: "(Number of favorable outcomes) ÷ (Total number of possible outcomes)."

Formula: \( P(A) = \frac{n(A)}{n(U)} \)
Note: Probability always stays between 0 and 1. If you get an answer like "120%," you’ve made a calculation error somewhere!

Solve "at least once" using Complementary Events!

This might be the most important technique in probability. When you see the words "at least once," think about the opposite scenario (never happens) and subtract it from the total (1).
\( P(A) = 1 - P(\bar{A}) \)

【Pro-tip!】
Whenever you see "at least," it’s a sign to "think about the opposite!"

5. Independent Trials and Conditional Probability

This part is a bit trickier and is frequently tested in the Common Test.

Independent Trials

This refers to cases where "the result of the first trial does not affect the second trial." For example, when rolling a die twice, the outcome of the first roll doesn't affect the second. In these cases, you simply multiply the probabilities.
\( P(A \cap B) = P(A) \times P(B) \)

Conditional Probability

This is "the probability that event B occurs, given that event A has already occurred."
Formula: \( P_A(B) = \frac{P(A \cap B)}{P(A)} \)

【Simple Way to Think】
Think of it as "the denominator (total universe) becoming smaller." For example, when considering "the probability that a card is a Heart, given that the card drawn is red," the denominator is not "all cards (52)" but is limited to "the red cards (26)."

Summary: The Key to Boosting Your Score

Finally, let’s summarize the study points for this chapter:

  • Distinguish between the "Addition Rule (+)" and the "Multiplication Rule (×)"!
  • Use \(P\) if order matters, and \(C\) if you are just choosing!
  • If you see "at least," subtract from 1 (Complementary Events)!
  • For conditional probability, be aware that "the denominator (the world) has changed!"

It might feel difficult at first, but you'll be fine. Start by working through the textbook examples repeatedly to get used to how these "tools" work. Once you start solving these puzzles, Math A will become much more fun! I'm rooting for you!