Calculating Simple Probabilities

Probability is a numerical measure that assigns a value to the likelihood of an event ($E$) occurring within a defined sample space ($S$). This measure quantifies how likely an event is to happen.

For many basic scenarios, especially those involving games of chance like flipping coins or rolling dice, we assume that all individual outcomes in the sample space are equally likely. A fair coin is just as likely to land heads as tails. A standard, fair die is equally likely to show any face from 1 to 6.

When outcomes are equally likely, calculating the probability of a specific event $E$ is straightforward. You count the number of outcomes that constitute the event $E$, and you divide that by the total number of possible outcomes in the sample space $S$.

The probability of an event $E$, denoted as $P(E)$, is calculated using the formula:

$$P(E) = \frac{\text{Number of favorable outcomes for event } E}{\text{Total number of possible outcomes in the sample space } S}$$

We can write this more compactly using set notation, where $|E|$ represents the number of elements (outcomes) in event $E$, and $|S|$ represents the total number of elements in the sample space $S$:

$$P(E) = \frac{|E|}{|S|}$$

Example: Rolling a Fair Six-Sided Die

Let's apply this. Consider the experiment of rolling a single, fair six-sided die.

1. Identify the Sample Space (S): The possible outcomes are the faces of the die. $S = \{1, 2, 3, 4, 5, 6\}$. The total number of possible outcomes is $|S| = 6$.

2. Define an Event (E): Let's say we're interested in the event $E$ of rolling an even number. The outcomes that satisfy this event are $E = \{2, 4, 6\}$. The number of outcomes favorable to event $E$ is $|E| = 3$.

3. Calculate the Probability: Using the formula:

   $$P(E) = \frac{|E|}{|S|} = \frac{3}{6} = \frac{1}{2}$$

   So, the probability of rolling an even number is 1/2 or 0.5 or 50%.

Let's try another event: What's the probability of rolling a number greater than 4?

• Event $F$: Rolling a number greater than 4.
• Favorable outcomes: $F = \{5, 6\}$.
• Number of favorable outcomes: $|F| = 2$.
• Total outcomes: $|S| = 6$.
• Probability: $P(F) = \frac{|F|}{|S|} = \frac{2}{6} = \frac{1}{3}$.

Example: Flipping a Fair Coin

Consider the simpler experiment of flipping a fair coin once.

1. Sample Space (S): $S = \{\text{Heads}, \text{Tails}\}$. Total outcomes $|S| = 2$.
2. Event (H): Getting Heads. $H = \{\text{Heads}\}$. Number of favorable outcomes $|H| = 1$.
3. Calculate Probability: $$P(H) = \frac{|H|}{|S|} = \frac{1}{2}$$ The probability of getting heads is 1/2 or 0.5.

Properties of Simple Probabilities

From the formula $P(E) = |E|/|S|$, we can see a couple of important properties:

1. Probability is always between 0 and 1: The number of favorable outcomes ($|E|$) can never be less than zero, and it can never be more than the total number of outcomes ($|S|$). Therefore:

   $$0 \le P(E) \le 1$$
   • A probability of 0 means the event is impossible (e.g., rolling a 7 on a standard six-sided die: $P(\text{rolling a 7}) = 0/6 = 0$).
   • A probability of 1 means the event is certain (e.g., rolling a number less than 10 on a standard six-sided die: $P(\text{rolling < 10}) = 6/6 = 1$).

2. Based on Equally Likely Outcomes: Remember, this simple formula relies heavily on the assumption that all individual outcomes in the sample space are equally likely. If you have a weighted die or a biased coin, this calculation method doesn't apply directly, and we need different approaches which we will touch upon later.

This method of calculating probabilities by counting favorable outcomes relative to the total number of outcomes forms the basis of classical probability and is a fundamental starting point for understanding more complex probability concepts.