【Math A】Number of Cases and Probability: Master Guide
Hello everyone! Welcome to the "Number of Cases and Probability" section of Math A.
You might be thinking, "This sounds kind of difficult..." or "There are too many formulas to memorize!" But don't worry!
The essence of this chapter is simply: "counting things carefully without gaps and without overlaps." Let's have fun and learn this like solving a puzzle!
1. Basics of Counting: Rule of Sum and Rule of Product
Before memorizing formulas, let's master the super important rule of "when to add and when to multiply."
(1) Rule of Sum (Add when you have "or"!)
When two events cannot happen at the same time, you calculate the number of outcomes by adding them.
Example: If you choose one item from "3 types of udon" or "2 types of soba" for lunch, there are \(3 + 2 = 5\) ways to choose.
(2) Rule of Product (Multiply when things happen "in succession" or "as a set"!)
When two events happen in sequence, or when you are considering them as a set, use multiplication.
Example: If you have 3 t-shirts and 2 pairs of pants, the number of outfits is \(3 \times 2 = 6\) ways.
【Pro-tip!】
If you're stuck, just ask yourself: "Can I choose them at the same time? (Addition)" or "Is this a set of choices? (Multiplication)"
2. Permutations: Order Matters!
We use permutations when we are "arranging" things. The key here is that "order matters."
(1) Basic Permutation \(nPr\)
The number of ways to choose \(r\) items from \(n\) items and arrange them in a row is given by:
\(nP_r = n(n-1)(n-2) \dots (n-r+1)\)
Example: Choosing the 1st, 2nd, and 3rd relay runners from 5 people:
\(5P_3 = 5 \times 4 \times 3 = 60\) ways
(2) Circular Permutation (Arranging in a circle)
When sitting at a round table, arrangements that are identical through rotation are considered the "same." Therefore, we fix one person in place and arrange the rest.
Formula: \((n - 1)!\)
【Fun Fact】
The "!" (factorial) means you multiply that number by every integer down to 1. For example, \(4! = 4 \times 3 \times 2 \times 1 = 24\). It looks just like an exclamation mark, doesn't it?
3. Combinations: Just Choosing is Enough!
Unlike permutations, with combinations, "order doesn't matter; you are just choosing."
(1) Basic Combination \(nCr\)
The number of ways to choose \(r\) items from \(n\) items is given by:
\(nC_r = \frac{nP_r}{r!} = \frac{n(n-1)\dots(n-r+1)}{r(r-1)\dots1}\)
Example: Choosing 3 people for cleaning duty from a group of 5 (there’s no order to the duty, right?):
\(5C_3 = \frac{5 \times 4 \times 3}{3 \times 2 \times 1} = 10\) ways
【Common Mistake】
Confusing "arranging" (Permutation \(P\)) with "just choosing" (Combination \(C\)) will drastically change your answer. If the problem uses words like "arrange," "in a row," or "position," use \(P\). If it just says "choose" or "form a group," suspect \(C\)!
★ A Handy Trick: \(nC_r = nC_{n-r}\)
Choosing 8 people out of 10 is hard work, but that is the same thing as choosing the 2 people who *won't* be picked.
In other words, \(10C_8 = 10C_2\), which makes your calculation much easier!
4. Basics of Probability: What is the Portion of the Whole?
Probability is found by: "(number of favorable outcomes) ÷ (total number of outcomes)."
\(P(A) = \frac{n(A)}{n(U)}\)
(1) Complementary Events (Use subtraction for "at least one"!)
Trying to calculate "at least one" directly can be a nightmare. In such cases, subtract the "cases that don't happen" from the whole (1).
Formula: \(P(A) = 1 - P(\bar{A})\)
Example: "Tossing a die twice, what is the probability that 1 appears at least once?"
→ Calculating "1 - (probability that 1 never appears in two tosses)" is way faster!
【Pro-tip!】
Whenever you see the phrase "at least one" in a problem, let a lightbulb go off: "Is this a complementary event (subtraction)?"
5. Independent Trials and Repeated Trials
This might feel a little tricky, but you'll be fine once you catch the pattern.
(1) Independent Trials
When outcomes do not affect each other (like tossing a die twice), you simply multiply their individual probabilities.
(2) Probability of Repeated Trials
This is for when you "repeat the same action many times."
If the probability of event A happening in one trial is \(p\), the probability that A occurs exactly \(r\) times in \(n\) trials is:
\(nC_r \times p^r \times (1-p)^{n-r}\)
Example: Probability of rolling a 1 exactly twice when tossing a die 5 times:
\(5C_2 \times (\frac{1}{6})^2 \times (\frac{5}{6})^3\)
It might seem hard at first, but just remember it as a set: "(in how many trials it happens: \(nC_r\)) × (probability of happening) × (probability of not happening)."
6. Conditional Probability (The "Zoom-In" Rule)
This is the "probability that something happens, given that we already know something else has occurred."
【Visualize it!】
Don't look at the "entire world." Instead, imagine "zooming your camera" only onto the people who fit the specific condition, and calculating the ratio within that group.
\(P_A(B) = \frac{n(A \cap B)}{n(A)}\)
Summary: How to Conquer This Chapter
1. Start by counting! Decide correctly between \(P\) and \(C\).
2. "At least one" means use a complementary event (subtract from 1).
3. For repeated trials, build the formula as a set (\(nC_r \cdot p \cdot q\)).
4. For conditional probability, restrict your denominator to the "range of the condition."
Don't worry at all if you make mistakes at first! By repeating mistakes like "Oh, I counted that twice" or "I forgot to account for the order," you will naturally develop the intuition. Take it one step at a time! I'm rooting for you!