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Understanding the probability of the union of two events is essential in determining the likelihood that at least one of two events will happen. The union of events A and B, denoted as \( A \bigcup B \), encompasses all outcomes that belong to either A, B, or both. The probability of this union is expressed as \( P(A \bigcup B) = P(A) + P(B) - P(A \bigcap B) \).

This formula essentially adds up the probabilities of each event occurring and then subtracts the probability of them happening simultaneously to account for overlapping outcomes, which would otherwise be double-counted. It's similar to how you would calculate the crowd at a concert that has fans of two bands performing. You count fans of band A, fans of band B, and then subtract the fans who like both to avoid counting them twice.

In our exercise, given \( P(A) = 0.5 \) and \( P(B) = 0.7 \), the assumed probability for the union was \( P(A \bigcup B) = 0.85 \). By applying these values to the formula, we can explore the relationship between A and B further.

The probability of the intersection of two events, \( A \bigcap B \), is fundamental when you want to find the probability that both events occur simultaneously. Imagine you're trying to find out how likely it is to roll a die and get an even number (event A) while also flipping a coin and getting heads (event B). The intersection is the occurrence of both an even die roll and the coin landing on heads.

To compute this, we already have the relationship \( P(A \bigcap B) = P(A) + P(B) - P(A \bigcup B) \). Using the values from our earlier equation, we deduced that \( P(A \bigcap B) = 0.35 \). This step is much like solving a puzzle by fitting the pieces of individual event probabilities together to see what they collectively reveal about the potential overlap.

When talking about conditional probability, you're dealing with the probability of an event A, given that another event B has already occurred. It is denoted by \( P(A | B) \), and it answers questions like, 'What are the chances of drawing an ace from a deck of cards given that we know the card is black?'

The conditional probability can be understood through the formula \( P(A | B) = \frac{P(A \bigcap B)}{P(B)} \) if B has a probability greater than zero. This equation essentially states that the probability of A happening, given B, is the probability of both A and B happening divided by the probability of B on its own.

In the context of independent events, if knowing that B occurs does not change the probability of A, then A and B are independent, and \( P(A | B) = P(A) \). However, in the exercise we saw that \( P(A \bigcap B) = 0.35 \) does not equal \( P(A) \times P(B) = 0.5 \times 0.7 \), indicating that A and B are not independent events. Their conditional probabilities would indeed differ, illustrating how these concepts interlink to portray a deeper understanding of event relationships.