As an expert in the field of probability and statistics, I can provide a comprehensive explanation of the "At Least One" rule, which is a concept that is particularly relevant when dealing with independent events.

The "At Least One" rule is a fundamental principle used in probability theory to calculate the likelihood of at least one event occurring within a set of independent events. To understand this rule, it's essential to first grasp the concept of independent events. Independent events are those where the occurrence of one event does not affect the probability of the occurrence of the other events. In other words, the outcome of one event is completely unrelated to the outcome of another.

When we talk about the "At Least One" rule, we are essentially looking at the probability of at least one event happening out of multiple independent events. This rule is particularly useful when we want to find the probability of a union of events, which is the probability that at least one of the events in a set will occur.

To calculate the probability of the "At Least One" rule, we can use the following formula:

\[ P(\text{At Least One}) = 1 - P(\text{None}) \]

Where \( P(\text{At Least One}) \) is the probability that at least one event occurs, and \( P(\text{None}) \) is the probability that none of the events occur.

The rationale behind this formula is that the probability of at least one event occurring is the complement of the probability that none of the events occur. In other words, the sum of the probabilities of these two complementary events must equal 1, or 100%.

Let's consider an example to illustrate this rule. Suppose we have a deck of cards, and we want to find the probability of drawing an ace from a five-card hand. There are four aces in a deck of 52 cards. The probability of drawing an ace on any given draw is \( \frac{4}{52} \), and the probability of not drawing an ace is \( \frac{48}{52} \). Since each draw is independent (assuming the cards are not replaced), we can calculate the probability of not drawing an ace in all five draws by multiplying the probabilities of not drawing an ace each time:

\[ P(\text{No Ace in 5 draws}) = \left(\frac{48}{52}\right)^5 \]

Now, to find the probability of drawing at least one ace, we apply the "At Least One" rule:

\[ P(\text{At Least One Ace}) = 1 - P(\text{No Ace in 5 draws}) \]

This calculation will give us the probability of drawing at least one ace in a five-card hand.

It's important to note that the "At Least One" rule is only applicable when the events are independent. If the events are not independent, the calculation would require a different approach, taking into account the conditional probabilities of the events.

In conclusion, the "At Least One" rule is a powerful tool in probability theory that allows us to determine the likelihood of at least one event occurring within a set of independent events. By understanding and applying this rule, we can solve a wide range of probability problems that involve independent events.

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Occasionally when calculating independent events, it is only important that the event occurs at least once. This is referred to as the 'At Least One' Rule. ... This means that the probability of the event never occurring and the probability of the event occurring at least once will equal one, or a 100% chance.read more >>