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Probability theory is a fundamental part of mathematics that deals with the likelihood of different outcomes. It provides tools to analyze random events and quantify the uncertainty. In the context of our example, we are dealing with selecting individuals from a group without replacement, meaning once an individual is selected, they are not put back into the pool.

When calculating probabilities, one needs to determine the total number of possible outcomes and then identify the number of favorable outcomes that meet the criteria of interest. This is essential for understanding the probability distribution, which is a list of all possible outcomes and their associated probabilities. In this instance, probabilities are calculated for scenarios where 0, 1, or 2 left-handed individuals are selected from the group.

Combinatorics is a branch of mathematics that studies counting, arrangement, and combination. It underpins the solution to our original exercise, where we calculated the total possible outcomes. Here, we used the combination formula, often noted as 'n choose r.'

This formula, denoted by \(nCr = \frac{n!}{(n - r)!r!}\), helps us determine how many ways we can choose \(r\) items from \(n\) total items, where order doesn't matter. In the original exercise, this was used to compute the number of ways to select 2 people from a pool of 12, and to separate different numbers of left-handed individuals from right-handed ones.

• Finding total possible combinations is computed by \(^{12}C_{2} = 66\).
• When choosing 2 right-handed people out of 9, the calculation is \(^{9}C_{2} = 36\).
• For selecting one from each group, left and right-handed, it's \(^{3}C_{1} \times ^{9}C_{1} = 27\).
• Finally, for both left-handed, use \(^{3}C_{2} = 3\).

Random sampling refers to the process of selecting individuals from a population in such a way that each individual has an equal chance of being chosen. However, the original problem involves sampling without replacement, meaning probability changes with each selection.

In simple terms, once a person is selected, they're not placed back into the pool, altering the probabilities of subsequent selections. This means each choice affects the pool's composition, and the probabilities for combinations have to account for these changes. It's important because it represents a real-world approach where selections often affect the pool, unlike with replacement where probabilities remain constant.

A discrete random variable is an important concept when dealing with probability distributions. It is a variable that can take on a countable number of distinct values, as part of an experiment or random process.

In our case, the random variable \(x\) represents the number of left-handed individuals chosen from the group. It's discrete because it can only take on specific values: 0, 1, or 2. Understanding this helps in constructing the probability distribution, which describes how probabilities are spread across possible values of the random variable.

• \(x = 0\): No left-handers selected, probability calculated as \(\frac{36}{66} = 0.545\).
• \(x = 1\): One left-hander selected, probability is \(\frac{27}{66} = 0.409\).
• \(x = 2\): Both selected are left-handed, probability is \(\frac{3}{66} = 0.045\).

These probabilities sum up to 1, confirming that they represent a full probability distribution for the variable \(x\).