2. Partial Differential Equations

A random experiment is an action or process that leads to one of several possible outcomes, where the result cannot be predicted with certainty beforehand.

Examples: • Tossing a coin • Rolling a die • Measuring the lifespan of a machine component

A random experiment is an action or activity where the outcome is uncertain. This means that when you perform the experiment, you can't know in advance what the result will be. For instance, if you toss a coin, it can land on either heads or tails, and you can't predict which one it will be. Similarly, when rolling a die, there are six possible outcomes, and any one of them can occur. This uncertainty in outcomes is a key characteristic of random experiments.

Think of a weather forecast. When meteorologists predict rain, they are essentially conducting a random experiment – they analyze various factors (like humidity, pressure, temperature) that can lead to rain, but the exact outcome (whether it will rain or not) is unpredictable until it happens.

The sample space is the set of all possible outcomes of a random experiment. • Notation: 𝑆 or Ω • It can be finite, countably infinite, or uncountably infinite.

✅ Examples: Experiment Sample Space Tossing a coin 𝑆 = {𝐻,𝑇} Rolling a die 𝑆 = {1,2,3,4,5,6} Tossing 2 coins 𝑆 = {𝐻𝐻,𝐻𝑇,𝑇𝐻,𝑇𝑇} Choosing a point in a square 𝑆 = {(𝑥,𝑦):0 ≤ 𝑥 ≤ 1,0 ≤ 𝑦 ≤ 1}

The sample space is a fundamental concept in probability. It represents all the possible outcomes of a random experiment. We denote it by the symbols S or Ω. The sample space can be of different types: it can have a finite number of outcomes (like tossing a coin, where the outcomes are heads or tails), countably infinite outcomes (like measuring the number of rolls of a die), or uncountably infinite outcomes (like choosing any point within a continuous range). Understanding the sample space helps you assess all possible results of an experiment.

Imagine you're at a candy store with various jars of candies. If you randomly pick a candy, the jars represent your sample space. Each jar has different types of candies, and when you pick one, you are drawing an outcome from various selections available in the jars. Just like the candy jars cover everything you could possibly choose (the outcomes), the sample space defines all possible outcomes of an experiment.

1. Discrete Sample Space: Contains a finite or countably infinite number of outcomes. 👉 Example: Rolling a die.
2. Continuous Sample Space: Contains uncountably infinite outcomes. 👉 Example: Measuring temperature in Celsius, say from 0°C to 100°C.

Sample spaces can be divided into two main types: discrete and continuous. Discrete sample spaces are those that have a limited number of outcomes, like rolling a die, where you can have 1, 2, 3, 4, 5, or 6. In contrast, continuous sample spaces have an infinite number of outcomes that cannot be counted, such as measuring temperature. In this case, you could have any temperature between 0°C and 100°C, including decimals like 0.1°C, 0.2°C and so on, leading to an uncountable number of outcomes.

Think about a game of lottery as a discrete sample space, where you can choose from specific numbers ranging from 1 to 50. You can only select one number at a time, making it a limited choice. On the other hand, when you're measuring your height, which can range anywhere from, say, 150 cm to 200 cm, you can have countless values in between (like 179.5 cm), representing a continuous sample space.

An event is a subset of the sample space. It can consist of one or more outcomes.

🧩 Types of Events:

Type | Description | Example

No detailed explanation available.

No real-life example available.