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Probability is a fundamental concept in statistics that measures the likelihood of an event occurring. In our exercise, we're interested in the probability regarding the height of a student chosen at random from the class. To calculate probability, we divide the number of favorable outcomes by the total number of possible outcomes. For instance, if we want to find the probability of selecting a student who is 5 feet 6 inches tall, we consider both the number of students in this height group and the total number of students in the class.

For this class:

• There are 3 students who are 5 feet 6 inches tall.
• The total number of students is 20.

Thus, the probability of this event is \(\frac{3}{20}\).Every height category can be calculated in a similar manner, allowing us to determine the probability for each height group. These probabilities serve as the weights when calculating the expected value, highlighting their importance in determining the likelihood and impact of each group's contribution to the class's average height.

A random variable is a numerical description of the outcome of a random event. In our context, the random variable represents the height of a student chosen at random. Each height category corresponds to a possible value of this random variable. When considering random variables, we account for:

• The values the random variable can take (in our case, different heights measured in inches).
• The probability associated with each of these values (the probability of each height being selected).

By converting each student's height into inches, we have created a common scale to work with our random variable. For example, 5 feet 6 inches becomes 66 inches. By understanding this random variable, we can move towards calculating an expected value, allowing us to predict the average result of the random event over multiple trials.

The weighted average is a crucial concept in calculating expected values, and it combines each outcome in the sample space with its probability. Here, the weighted average allows us to determine the average height of a student by considering both their individual likelihood of being chosen and their corresponding height.To compute the weighted average, we:

• Multiply each possible height by its corresponding probability.
• Sum all these products.

For our problem, this means:\[E(X) = \left(\frac{3}{20} \times 66\right) + \left(\frac{5}{20} \times 68\right) + \left(\frac{4}{20} \times 70\right) + \left(\frac{4}{20} \times 72\right) + \left(\frac{4}{20} \times 74\right)\]Each term in this sum reflects the weighted contribution of that height category to the overall expected height. By performing these calculations, we find the expected height, which in this case is 70.1 inches. This weighted average provides us with a practical understanding of what height to expect if we repeatedly selected students at random from the class.