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Probability theory is the branch of mathematics that deals with the analysis of random phenomena. It provides the foundation for understanding events, their likelihoods, and how they interrelate. At its core are basic definitions:

• **Experiment**: Any process or action that can produce a set of outcomes.
• **Event**: A specific outcome or a set of outcomes of an experiment.
• **Probability**: A measure, between 0 and 1, of the likelihood that an event will occur.

In probability calculations, events can be classified in various ways, one way being to identify them as either dependent or independent events. This distinction is very important as it determines how probabilities are calculated and interpreted.

Dependent events are those events where the occurrence of one event affects the likelihood of the occurrence of the other event. In other words, the probability of one event changes when we know that another event has occurred. To understand dependent events better, we can look at part (a) from the exercise. Here, **E** is 'It rains on June 30' and **F** is 'It is cloudy on June 30'. These events are dependent because raining usually necessitates the presence of clouds.
Another example is in part (c): **E** is 'You live at least 80 years', and **F** is 'You smoke a pack of cigarettes every day of your life'. These events are also dependent. Smoking is known to reduce lifespan, thus influencing the likelihood of living to 80 years.

Independent events are events where the occurrence of one event does not affect the occurrence of the other. For events **E** and **F** to be independent, the equation \( P(E \cap F) = P(E) \cdot P(F) \) must hold true. This means the joint probability of both events occurring is the product of their individual probabilities. Looking at part (b) from the exercise:

• **E**: 'Your car has a flat tire'
• **F**: 'The price of gasoline increases overnight'

These events are independent. The probability of getting a flat tire has no impact on the probability of gas prices changing, so their joint probability equals the product of their individual probabilities.

Probability calculations involve determining the likelihood of events occurring. For dependent events, you need to consider the conditional probabilities:
\[ P(E \cap F) = P(E | F) \cdot P(F) \] On the other hand, for independent events, the calculation simplifies to:
\[ P(E \cap F) = P(E) \cdot P(F) \]
Let's revisit the exercise:

• In part (a), where the probability of rain impacts the probability of it being cloudy, we calculate these events taking dependency into account.
• In part (b), the occurrences are unrelated. Therefore, calculating their joint probability is straightforward with \( P(E \cap F) = P(E) \cdot P(F) \).
• Part (c) is another scenario of dependent events, where smoking influences the probability of living longer.

Understanding whether events are independent or dependent is crucial in calculating probabilities accurately.