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Probability theory is a foundational element in statistics and mathematics. It gives us a systematic way to determine how likely events are to happen. When we talk about probability, we often express it as a fraction or percentage that ranges between 0 and 1, where 0 means an event will not happen, and 1 means it will definitely happen.

To calculate probability, we use the formula \( P = \frac{\text{Number of favourable outcomes}}{\text{Total number of outcomes}} \). This means we look at how many outcomes meet our criteria versus the total number of possibilities.

• *Total number of outcomes* refers to all the possible outcomes that can occur.
• *Favourable outcomes* are the outcomes that align with the event we are interested in.

In many practical problems, identifying these outcomes is the first step in solving probability-related questions.

Favourable outcomes are crucial when calculating probabilities. They are the specific outcomes of an event that fulfill the condition we are interested in. For examples like rolling a dice, it's important to recognize which numbers represent the outcomes you are focusing on.

In the example from the exercise, the condition was rolling a number greater than 7 with a standard die. As a standard die has numbers from 1 to 6, there are no outcomes that meet the required condition. This means:

• There are 0 favourable outcomes that satisfy the condition (number greater than 7).

If there are no favourable outcomes, the probability of the event is zero, as an event with no favourable circumstances cannot occur.

Dice probability is a common topic when discussing probability due to the simplicity and familiarity of dice games. A standard die has six faces, each marked with a distinct number from 1 to 6. This makes calculating probabilities with a die straightforward.

For any single roll of the die, there are 6 possible outcomes. Each face is equally likely to show up, assuming the die is fair. When determining the probability of events with a die, you follow these steps:

• Define the event: What are you looking to observe?
• Identify the favourable outcomes among the 6 faces.
• Use the probability formula \( P = \frac{\text{Number of favourable outcomes}}{6} \).

For instance, the probability of rolling a 3 is \( \frac{1}{6} \), because there is one 3 on a six-faced die. Yet, if the event was to roll a number greater than 7, it would be \( \frac{0}{6} = 0 \), since such an outcome is impossible.