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When considering the probability of outcomes in a lottery draw, it's essential to understand that the drawing of each ball is considered an independent event. In probability theory, independent events are those whose outcomes do not affect one another. This concept is crucial for calculating probabilities because the likelihood of each event must be evaluated separately.

For the New York state lottery, each of the three bins holds 10 balls numbered from 0 to 9. The selection of a ball from one bin does not influence the selection from another. Therefore, the chance of drawing any specific number from each bin remains constant. In our example, to find the probability of the sequence 9-1-1, you calculate the chance of drawing a '9' followed by a '1', and another '1', multiplying these independent probabilities. Thus, understanding independent events helps in finding the probability of more complex sequences by simply multiplying separate probabilities together.

Classical probability is the method used in part (a) of the given exercise. This approach assumes that each outcome in a probability experiment is equally likely to occur. It is particularly useful when dealing with finite sample spaces, such as the selection of lottery numbers.

In our scenario, because each bin contains balls numbered 0 through 9, there are 10 possible outcomes for each ball selection. As one ball is drawn from each bin independently, and assuming each draw is fair and random, all sequences should have an equal chance of being selected. Here, the formula for classical probability is applied, which states that the probability of a particular outcome can be calculated by dividing the number of favorable outcomes by the total number of possible outcomes. With 10 possible numbers per bin, the chance to get '911' becomes the product of independent probabilities: \[\frac{1}{10} \times \frac{1}{10} \times \frac{1}{10} = 0.001\].

This approach to probability is grounded in theoretical expectations, providing a straightforward and logical way to understand and calculate probabilities.

Lottery probability refers to the calculation of the chances of selecting a specific sequence in a lottery setting, such as the one featured in the New York state lottery. These lotteries often operate by having separate bins or drums, each containing numbered balls, from which one ball is drawn at a time.

The goal in these settings is to determine the probability of drawing a specified sequence of numbers, like 9-1-1. In this type of lottery, since each bin operates independently and each ball has an equal chance of being selected, the computation of such probability requirements is straightforward. The probability of any particular number being drawn from each bin is the same, simplifying the multiplication of probabilities across independent draws.

Despite the apparent complexity of lotteries, the underlying principle of probability in these settings often relies on the concept of multiplying simple probabilities of independent events. This ensures that each number combination retains an equal playing field, reflecting true randomness.

Theoretical probability involves calculating the expected outcomes based on a fully defined model or scenario. It is a method of determining probabilities which does not rely on experimental data but on the inherent logic of the situation.

In the context of the exercise, we used theoretical probability to find the likelihood of the 9-1-1 sequence appearing. We based our calculations on the assumption that each number was equally likely to be chosen, a premise backed by the setup of the lottery: bins filled with numbers 0 to 9. This assumption meant that each draw was fair and random, allowing us to calculate the probability of the sequence without needing past data.

Theoretical probability stems from an understanding of the mechanics of the probability setup, such as the independence of events and equal likelihood of results. It provides a logical foundation from which predictions can be derived, largely based on mathematical reasoning and principles.