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In the realm of probability, a random variable is a fundamental concept that represents a quantity whose value is subject to varying outcomes of a random phenomenon. Imagine tossing a coin; the result can be either head or tail. The random variable in this scenario could represent the number of heads you get in a series of tosses. The values a random variable can take are associated with different probabilities.

In the exercise, the variable 'x' is a discrete random variable since it has specific separate values it can take, such as 2, 2.5, 3.0, and so on. The table is essentially a probability distribution that tells us how probable each outcome is. Understanding this helps you accurately determine the likelihood of any given occurrence of 'x'.

The expected value is a measure that gives us an idea of the 'average' outcome we can anticipate from a random variable over many trials. It takes into account all the possible values of the random variable and their respective probabilities. To calculate it, we multiply each possible value of the variable by its probability and sum up these products.

For example, if a dice is rolled, the expected value of the outcome would be calculated as: \( (1 \times \frac{1}{6}) + (2 \times \frac{1}{6}) + (3 \times \frac{1}{6}) + (4 \times \frac{1}{6}) + (5 \times \frac{1}{6}) + (6 \times \frac{1}{6}) \), which equals 3.5.

In our exercise, to find the expected number of times 'x' will equal 2.5 when sampled 50000 times, we use the provided probability of 0.36 and multiply it by 50000, which gives us an expected count of 18000 times.

Probability calculations are used to quantify the likelihood of events represented by a random variable using a scale from 0 to 1, where 0 denotes impossibility and 1 denotes certainty. To solve various probability questions for a discrete random variable, you sum the probabilities of the outcomes that satisfy the condition in question.

For instance, for part (a) 'P(x=3.5)', one simply looks up the value directly in the distribution table. In more compound situations, like calculating 'P(x \geqslant 3.0)', you need to add the probabilities for all 'x' values that are equal to or greater than 3.0.

It's important to read the inequalities correctly: 'greater than or equal to' (\( \geqslant \) ) and 'less than' (\( < \)) indicate which values to include in your calculation. The sums of these probabilities provide answers to the posed queries regarding the behavior of the random variable 'x'.