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Independent events in probability are scenarios where the result of one event does not influence or change the result of another. To determine if events are independent, check if knowing the outcome of one event gives no information about the outcome of the other.
For example, let's take the events from the coin toss exercise. Event A is getting at least one head in two tosses, and Event B is getting two tails. These events are not independent. This is because if you know event A occurred (you have at least one head), event B cannot occur as it requires both coin tosses to be tails.
In essence, two events are independent if and only if the probability of both happening is the product of their individual probabilities:

• \(P(A \text{ and } B) = P(A) \times P(B)\)

If this condition does not hold, then the events are dependent, as in our example.

Complementary events are a pair of outcomes where one outcome or event happening means the other cannot happen. When summed, their probabilities equal 1. One classic example is the chance of rain versus no rain on a specific day.
In the given exercise, we have events A and B from a coin toss. Event A is getting at least one head, while event B is getting two tails. Importantly, these are not complementary events. A complementary event would mean that the probability of event A happening plus the probability of event B happening equals 1.
Since there is an outcome (HH) where neither event A nor event B occurs, these events don't fit the definition of complementary events. Therefore, we can't use the formula

• \(P(A) = 1 - P(B)\)

for these events.

Calculating probabilities involves straightforward mathematics. It requires knowing the number of favorable outcomes and the total outcomes possible. For coin tosses, each toss is independent, but the total outcome space remains four combinations: {HH, HT, TH, TT}.
Here are the step-by-step calculations for the example:- **Event B (Two tails):** There's only one outcome where both tosses result in tails, TT. Therefore, the probability of event B is:

• \(P(B) = \frac{1}{4}\)

- **Event A (At least one head):** This can occur in three out of the four possible outcomes: HH, HT, and TH. Therefore, the probability of event A is:

• \(P(A) = \frac{3}{4}\)

Unlike complementary events, we calculate these probabilities separately, considering all possible outcomes for each event.