Probability - The Outcome Table

Probability tables, outcome spaces, and sample spaces are concepts commonly used in probability theory to describe and analyze random experiments and events. Let's break down each term and provide examples:

1. Sample Space:

   • Definition: The sample space, denoted as $S$, is the set of all possible outcomes of a random experiment.
   • Example: Consider rolling a six-sided die. The sample space for this experiment is $S = \{1, 2, 3, 4, 5, 6\}$.

2. Outcome Space:

   • Definition: The outcome space is a subset of the sample space that represents the outcomes associated with a particular event.
   • Example: If the event is getting an even number when rolling a six-sided die, the outcome space is $E = \{2, 4, 6\}$, which is a subset of the sample space.

3. Probability Table:

   • Definition: A probability table lists all possible outcomes of an experiment along with their associated probabilities.

   • Example: Let's say we're rolling a fair six-sided die, and each outcome has an equal probability of $\frac{1}{6}$. The probability table would look like this:

     Outcome	Probability
     1	1/6
     2	1/6
     3	1/6
     4	1/6
     5	1/6
     6	1/6

   This table shows that each outcome has an equal probability of $\frac{1}{6}$.

Sample Working:

Let's calculate the probability of getting an even number when rolling a fair six-sided die using the concepts mentioned above.

1. Sample Space (S): $S = \{1, 2, 3, 4, 5, 6\}$

2. Outcome Space for Even Numbers (E): $E = \{2, 4, 6\}$

3. Probability Table:

4. Calculate Probability: The probability of getting an even number is the sum of the probabilities associated with each even outcome.

   $$P(\text{Even}) = P(2) + P(4) + P(6) = \frac{1}{6} + \frac{1}{6} + \frac{1}{6} = \frac{3}{6} = \frac{1}{2}$$

So, the probability of rolling an even number is $\frac{1}{2}$.