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To understand mutually exclusive events, think of situations where two outcomes cannot happen at the same time. Imagine you are flipping a coin; you can't get a head and a tail simultaneously. Therefore, for mutually exclusive events, the probability that both occur together, noted as \( P(A \cap B) \), is zero. In the exercise, when events are mutually exclusive, \( P(A \cap B) = 0 \).
When calculating the probability of either event occurring, called the union \( P(A \cup B) \), use the formula:

• \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \).

Since \( P(A \cap B) \) is zero for mutually exclusive events, the union simplifies to just \( P(A) + P(B) \). This is evident in Step 2 of the solution, where the union of 0.3 and 0.4 results in 0.7.
Additionally, the conditional probability \( P(A | B) \) describes the likelihood of event \( A \) given \( B \) has already happened. For mutually exclusive events, it is zero because the occurrence of \( B \) prevents \( A \). Hence, in Step 3, \( P(A | B) = 0 \).

Independent events occur when the outcome of one does not affect the other. A common example is rolling a die and flipping a coin; one doesn't influence the other.
Mathematically, two events \( A \) and \( B \) are independent if and only if \( P(A \cap B) = P(A) \times P(B) \). Step 4 demonstrates this, where \( P(A \cap B) \) is calculated as 0.3 times 0.4, equating to 0.12.
To find the probability of either event occurring, or their union \( P(A \cup B) \), the equation remains the same:

• \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \).

For independent events in Step 5, \( P(A \cup B) \) is 0.58.
Conditional probability \( P(A | B) \), which tells us the probability of \( A \) given that \( B \) has occurred, is given by \( \frac{P(A \cap B)}{P(B)} \). Step 6 shows this calculation to be 0.3, confirming the independence as the conditional probability equals \( P(A) \) itself.

Conditional probability allows us to find the probability of event \( A \) happening, assuming event \( B \) has already occurred. This is expressed as \( P(A | B) \). The formula used is:

• \( P(A | B) = \frac{P(A \cap B)}{P(B)} \).

This formula is essentially about narrowing down the sample space to just where \( B \) happens, giving insights about \( A \) in this specific scenario.
For example, in Step 10 when \( P(A) = 0.2 \) and \( P(B) = 0.5 \), given the intersection \( P(A \cap B) = 0.1 \), it reveals that \( P(A | B) = 0.2 \). It shows how specific details about one event alter our understanding of the probability of another.
In practice, pay attention to whether events are independent, mutually exclusive, or conditional before applying formulas. This clarifies which probabilities remain consistent and which need specification of conditions, thereby sharpening the precision of your predictions and analyses in probability theory.