🌟 Level 2 - Topic 9: Introduction to Probability & Combinatorics (Part 1) 🎲🔢

1) Introduction to Probability & Combinatorics 🎲🔢

Welcome to a new and exciting area of mathematics: Probability and Combinatorics! So far in Level 2, we've explored sequences and series, focusing on patterns of numbers. Now, we're shifting gears to explore the mathematics of chance and counting.

From predicting weather patterns to understanding the odds in games of chance, probability is a powerful tool in making decisions and understanding the world around us.

Combinatorics is essential in many areas, including computer science (algorithm analysis), statistics (experimental design), and probability itself (calculating favorable outcomes).

In this Part 1, we'll take our first steps into both of these fascinating fields. We will focus on:

• Understanding the basic concepts of probability: experiments, outcomes, sample spaces, and events.
• Learning how to calculate basic probabilities when outcomes are equally likely.
• Exploring the fundamental Basic Counting Principle (also known as the Multiplication Principle) from combinatorics.

Get ready to unlock the power of chance and counting! Let's begin our journey into Probability & Combinatorics! 🚀

2) Introduction to Probability 🎲

2.1 Basic Probability Concepts

Probability is a measure of how likely an event is to occur. It is always expressed as a number between 0 and 1, inclusive. A probability of 0 means the event is impossible, while a probability of 1 means the event is certain. Probabilities closer to 1 indicate higher likelihood.

To understand probability, we need to define some key terms:

• Experiment (or Trial): An experiment is any process or action that can be repeated and has well-defined possible results, called outcomes. Examples: tossing a coin, rolling a die, drawing a card from a deck.
• Outcome: An outcome is a single possible result of an experiment. For example, when tossing a coin, the possible outcomes are "Heads" or "Tails".
• Sample Space: The sample space, often denoted by \(S\), is the set of all possible outcomes of an experiment. For a die roll, the sample space is \(S = \{1, 2, 3, 4, 5, 6\}\). For a coin toss, \(S = \{\text{Heads, Tails}\}\).
• Event: An event is any subset of the sample space. It's a specific outcome or a collection of outcomes that we are interested in. For example, in a die roll, the event "rolling an even number" corresponds to the set of outcomes \( \{2, 4, 6\} \).

2.2 Calculating Basic Probability

In situations where all outcomes in the sample space are equally likely to occur, we can calculate the probability of an event using a simple formula:

\( P(\text{Event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \)

Here, "favorable outcomes" are the outcomes that are part of the event we are interested in. "Total number of possible outcomes" is the total number of outcomes in the sample space.

It's important to remember that this basic formula applies when outcomes are equally likely. For example, when using a fair coin or a fair die.

The probability value \(P(\text{Event})\) will always be between 0 and 1, inclusive: \( 0 \leq P(\text{Event}) \leq 1 \). It can also be expressed as a percentage (by multiplying by 100%).

Example 1: Probability of Rolling an Even Number on a Die

What is the probability of rolling an even number when you roll a fair six-sided die?

**Solution:**

• Experiment: Rolling a die.
• Sample Space: \(S = \{1, 2, 3, 4, 5, 6\}\). Total number of possible outcomes = 6.
• Event: "Rolling an even number". Favorable outcomes are \( \{2, 4, 6\} \). Number of favorable outcomes = 3.

Using the formula: \( P(\text{Event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \)

\( P(\text{Even number}) = \frac{3}{6} = \frac{1}{2} \)

The probability of rolling an even number is \( \frac{1}{2} \) or 50%.

Example 2: Probability of Drawing a Heart from a Deck of Cards

What is the probability of drawing a heart when you randomly draw one card from a standard deck of 52 playing cards?

**Solution:**

• Experiment: Drawing a card from a deck.
• Sample Space: A standard deck of 52 cards. Total number of possible outcomes = 52.
• Event: "Drawing a heart". There are 13 hearts in a deck. Number of favorable outcomes = 13.

Using the formula: \( P(\text{Event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \)

\( P(\text{Drawing a heart}) = \frac{13}{52} = \frac{1}{4} \)

The probability of drawing a heart is \( \frac{1}{4} \) or 25%.

3) Introduction to Combinatorics: The Basic Counting Principle 🔢

3.1 The Basic Counting Principle (Multiplication Principle)

Combinatorics is all about counting, and one of the most fundamental tools in combinatorics is the Basic Counting Principle, also known as the Multiplication Principle. This principle is used to find the total number of outcomes when you have a sequence of events.

In simpler terms, if you have multiple choices to make, and the number of choices for each step is independent of the previous choices, you multiply the number of choices at each step to get the total number of combined possibilities.

Example 3: Counting Outfits

Suppose you have 3 shirts, 2 pairs of pants, and 4 pairs of shoes. How many different outfits can you create if an outfit consists of one shirt, one pair of pants, and one pair of shoes?

**Solution:**

• Number of choices for shirts (\(n_1\)) = 3
• Number of choices for pants (\(n_2\)) = 2
• Number of choices for shoes (\(n_3\)) = 4

Using the Basic Counting Principle, the total number of outfits is:

Total outfits = \(n_1 \times n_2 \times n_3 = 3 \times 2 \times 4 = 24\)

You can create 24 different outfits.

Example 4: Counting Possible License Plates

A license plate consists of 3 letters followed by 4 digits. How many different license plates are possible if letters and digits can be repeated?

**Solution:**

• Choices for the 1st letter = 26 (assuming English alphabet)
• Choices for the 2nd letter = 26 (repetition allowed)
• Choices for the 3rd letter = 26 (repetition allowed)
• Choices for the 1st digit = 10 (0-9)
• Choices for the 2nd digit = 10 (repetition allowed)
• Choices for the 3rd digit = 10 (repetition allowed)
• Choices for the 4th digit = 10 (repetition allowed)

Using the Basic Counting Principle, the total number of license plates is:

Total license plates = \(26 \times 26 \times 26 \times 10 \times 10 \times 10 \times 10 = 26^3 \times 10^4 = 175,760,000\)

There are a huge number – over 175 million – possible license plates! This principle allows us to count possibilities even in complex scenarios.

4) Examples (Detailed) 🍀

Example 5: Probability with Dice and Cards Combined

Let's combine probability and combinatorics in a slightly more involved example.

1. Challenge 1: What is the probability of rolling a sum of 7 when you roll two fair six-sided dice?

2. Challenge 2: What is the probability of drawing a King AND then drawing a Queen (without replacement) from a standard deck of 52 cards?

**Let's solve each challenge:**

1. Solution to Challenge 1: Probability of Sum 7 with Two Dice

   First, we need to determine the total number of possible outcomes when rolling two dice. For each die, there are 6 outcomes. Using the Basic Counting Principle, the total number of outcomes is \(6 \times 6 = 36\). These outcomes are equally likely (if dice are fair).

   Now, we need to find the number of favorable outcomes – pairs that sum to 7. Let's list them as pairs (die 1, die 2):

   \( \{(1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)\} \)

   There are 6 favorable outcomes.

   Therefore, the probability of rolling a sum of 7 is:

   \( P(\text{Sum of 7}) = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{6}{36} = \frac{1}{6} \)

2. Solution to Challenge 2: Probability of King then Queen (without replacement)

   Here, we have two events happening in sequence: drawing a King first, then a Queen (without putting the King back).

   Step 1: Probability of drawing a King first. In a deck of 52 cards, there are 4 Kings. So, the probability of drawing a King as the first card is \( \frac{4}{52} \).

   Step 2: Probability of drawing a Queen second (given a King was drawn first). After drawing a King and NOT replacing it, there are now only 51 cards left in the deck. Among these 51 cards, there are still 4 Queens (since we only removed a King). So, the probability of drawing a Queen as the second card, given that a King was drawn first, is \( \frac{4}{51} \).

   To find the probability of both events happening in sequence, we multiply their probabilities:

   \( P(\text{King then Queen}) = P(\text{King first}) \times P(\text{Queen second | King first}) \)
   \( P(\text{King then Queen}) = \frac{4}{52} \times \frac{4}{51} = \frac{16}{2652} = \frac{4}{663} \) (approximately \( \frac{1}{165.75} \))

   The probability of drawing a King and then a Queen (without replacement) is \( \frac{4}{663} \), which is quite small.

5) Practice Questions 🎯

6) Summary 🎉

• Probability is the study of chance and uncertainty, quantifying the likelihood of events.
• Key probability concepts include: Experiment, Outcome, Sample Space, and Event.
• For equally likely outcomes, probability is calculated as: \( P(\text{Event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \).
• Combinatorics is the mathematics of counting.
• The Basic Counting Principle (Multiplication Principle) states that to find the total number of outcomes of a sequence of tasks, you multiply the number of ways to perform each task.
• Probability values are always between 0 and 1.

Congratulations on taking your first steps into the world of Probability and Combinatorics! You've learned fundamental concepts of probability and the Basic Counting Principle. These are powerful tools for understanding chance and counting possibilities. In Part 2, we will expand our combinatorics toolkit and explore more probability scenarios! Keep practicing, and you'll become a master of chance and counting! 🌟