2.1.2. Sample Space (S)

The sample space is the set of all possible outcomes of a random experiment.

The sample space is a critical concept in probability theory that refers to the collection of all potential outcomes from a given random experiment. For example, if we consider the random experiment of rolling a die, the sample space includes all the possible results you could get when the die is rolled. Every outcome is important, as they form the basis from which probabilities can be calculated.

Imagine you're baking a cake and you have three potential types of frosting to choose from: chocolate, vanilla, and strawberry. The sample space in this scenario would represent all your choices for frosting. Just like you need to consider all frosting options for your cake, in probability, we need to consider all possible outcomes to understand the full picture.

• Notation: 𝑆 or Ω

In the field of probability, sample spaces are often represented using specific symbols. The common notations for sample space are 'S' or 'Ω' (the Greek letter Omega). These symbols serve as concise representations to indicate the set of all possible outcomes in discussions and calculations involving probabilities.

Think of the notation for sample space like a label on a box of assorted chocolates. The label tells you that all the chocolate varieties are inside, just as 'S' or 'Ω' tells us that all possible outcomes are included in that 'box' of probability.

• It can be finite, countably infinite, or uncountably infinite.

Sample spaces can be categorized based on the number of outcomes they contain. A finite sample space has a limited number of possible outcomes, like the colors of marbles in a jar. A countably infinite sample space has an infinite number of outcomes that can be enumerated, such as the natural numbers. An uncountably infinite sample space has outcomes that cannot be counted one-by-one, such as the points within a given range on a line.

Consider a jar filled with 10 different types of marbles. The sample space here is finite because you can count them all. Now think about a number line that includes all real numbers between 0 and 1. This space is uncountably infinite because between any two numbers, you can always find another number. Each type of outcome helps us understand different kinds of experiments in probability.

✅ Examples:

• 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}

Here are some practical examples of sample spaces for different experiments: 1) When tossing a coin, the sample space consists of two outcomes - heads (H) and tails (T). 2) In the case of rolling a standard six-sided die, the sample space consists of six outcomes: {1, 2, 3, 4, 5, 6}. 3) When tossing two coins, the sample space is expanded, with four outcomes: combinations of heads and tails. 4) Finally, choosing a point in a square introduces a continuous sample space, where any point within the region defined by 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1 is included.

Consider tossing a coin; you can only get heads or tails, similar to making a simple choice between two options. Rolling a die introduces more choices, just like deciding on an activity with six different options to pick from. Tossing two coins makes the decision-making process even more interesting—like having multiple friends over, where the outcomes reflect their preferences! Lastly, choosing a point in a square can be thought of like picking a random spot in a park – there are countless possibilities!

--