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In probability and statistics, a discrete random variable is a type of random variable that can take a finite or countable number of distinct values. Think of it as counting outcomes like the toss of a die or the number of muffins sold, as in our baker's scenario.
A discrete random variable has an associated probability distribution which lists each of these possible values and their corresponding probabilities. It's common to see this represented in a table that shows these probabilities alongside each possible value of the random variable, just like in the baker's problem where each batch quantity has a specific probability.
Important characteristics to note:

• Each probability must be between 0 and 1.
• The sum of all probabilities must equal 1.

These rules are crucial since they ensure that one of the outcomes will definitely happen. In the case of the baker, the distribution reflects the different likely sales scenarios he faces when he decides how many batches of muffins to bake.

When calculating probabilities associated with a discrete random variable, you need to focus on the relevant probabilities for the specific outcomes you are interested in. For example, the question about the muffins asks us to find the probability that more than one batch is sold.
This means we calculate the probability of selling 2, 3, or 4 batches, which are the values greater than 1. The task is to sum up the probabilities associated with these outcomes.
Here's how you do it:

• Find each relevant probability from the table. In this case, it's 0.35 for 2 batches, 0.40 for 3 batches, and 0.10 for 4 batches.
• Add these probabilities to find the total probability for sales greater than one batch: \[P(x > 1) = P(x = 2) + P(x = 3) + P(x = 4) = 0.35 + 0.40 + 0.10 = 0.85.\]

This calculation is simple, but correctness is crucial as it determines your understanding of the distribution.

Applied statistics use statistical methods and theories to solve real-world problems. The baker's decision-making process regarding muffins is an excellent example of applied statistics in action. Here, he's using known probabilities to make educated guesses about his daily sales, ensuring he makes just the right number of muffins.
This involves:

• Collecting data, such as past sales records, which helps establish a probability distribution.
• Analyzing the data to understand the likelihood of different scenarios.
• Making decisions based on data-driven insights.

The baker uses statistical analysis to minimize waste while maximizing sales, demonstrating the practical application of statistics. Through these efforts, decisions aren't just based on intuition but backed by data, providing clarity and reducing financial risk.