Theoretical Possibility Spaces for Probability

Introduction to Theoretical Probability

Probability is the mathematical measure of how likely an event is to occur. In the context of the TMUA and ESAT, we focus on theoretical probability, which is determined by analysing the physical or logical properties of an experiment rather than performing repeated trials. This analysis assumes that certain outcomes are equally likely. For example, when rolling a fair six sided die, we assume each number from 1 to 6 has the same chance of appearing.

Constructing Possibility Spaces

A possibility space, often called a sample space, is a comprehensive list or diagram showing every possible outcome of an experiment. To calculate probability correctly, the outcomes listed in your space must be mutually exclusive, meaning no two outcomes can happen at once, and exhaustive, meaning they cover every possibility. We use the symbol $S$ to represent the sample space and $n(S)$ to represent the total count of outcomes in that space.

Single Experiments

For a single experiment, the possibility space is often a simple list. For a fair coin, the space is $S = \{H, T\}$, where $H$ denotes heads and $T$ denotes tails. For a standard die, $S = \{1, 2, 3, 4, 5, 6\}$. In these cases, $n(S) = 2$ and $n(S) = 6$ respectively. The probability of any single outcome occurring is $\frac{1}{n(S)}$.

Combined Experiments and Sample Space Diagrams

When two or more experiments are performed together, such as rolling two dice or flipping three coins, constructing the possibility space requires a systematic approach to ensure no outcomes are missed or counted twice.

Two Way Grids

A two way grid, or sample space diagram, is the most effective way to represent the outcomes of two combined experiments. For example, if we roll two fair six sided dice, we can construct a grid where the rows represent the outcomes of the first die and the columns represent the outcomes of the second die.

This results in a 6 by 6 grid containing 36 unique outcomes. Each cell in the grid represents a pair of results, such as $(1, 1)$, $(1, 2)$, through to $(6, 6)$. Because the dice are independent and fair, every one of these 36 outcomes is equally likely.

Worked Example: The Sum of Two Dice

Consider rolling two fair dice and finding the probability that the sum of the scores is 9. First, we identify the total number of outcomes in the sample space, which is $n(S) = 36$. Next, we list the successful outcomes where the sum equals 9:

1. Die 1 is 3 and Die 2 is 6: $(3, 6)$
2. Die 1 is 4 and Die 2 is 5: $(4, 5)$
3. Die 1 is 5 and Die 2 is 4: $(5, 4)$
4. Die 1 is 6 and Die 2 is 3: $(6, 3)$

There are $n(E) = 4$ successful outcomes. The theoretical probability is calculated as:

$P(\text{Sum is } 9) = \frac{n(E)}{n(S)} = \frac{4}{36} = \frac{1}{9}$

Systematic Listing for Multi Stage Experiments

For experiments involving three or more stages, such as flipping three coins, a grid is not possible. Instead, we use systematic listing or tree diagrams. To list outcomes systematically, keep one variable constant while changing others. For three coin flips, the sample space is:

$S = \{HHH, HHT, HTH, THH, HTT, THT, TTH, TTT\}$

Here, $n(S) = 8$. Each outcome, such as $HHT$, is distinct from $HTH$. This distinction is vital because each outcome represents a different path through the experiment, and every path is equally likely.

Worked Example: Three Coins

Find the probability of obtaining exactly two heads when flipping three fair coins. From our listed sample space, we identify the outcomes with exactly two heads: $\{HHT, HTH, THH\}$. There are 3 such outcomes. Thus:

$P(\text{Exactly 2 heads}) = \frac{3}{8}$

Theoretical Probability with Tree Diagrams

Tree diagrams are another visual tool for constructing possibility spaces. Each branch represents a possible outcome of a single stage. By following every path from the start to the end of the tree, you can list the complete sample space. If each individual choice at every junction is equally likely, then every final path at the end of the tree is also equally likely. To calculate the probability, count the number of paths that satisfy your criteria and divide by the total number of paths.