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In probability theory, a random variable is a way to quantify and understand how likely various outcomes of an event are. Here, random variables are used to model the lateness of people attending a seminar. Think of a random variable as a function that assigns numerical values to the outcomes of a random phenomenon.

In our scenario, each couple or individual invited to an investment seminar is associated with a binary random variable. For couples, if they are late, the random variable is 1, otherwise, it's 0. Likewise, for each individual. We denote these random variables as \( C_i \) for couples and \( I_i \) for single individuals, where \( i \) is the index.

• \( C_i = 1 \) if the \(i\)-th couple is late (together).
• \( C_i = 0 \) if the \(i\)-th couple is on time.
• \( I_i = 1 \) if the \(i\)-th individual is late.
• \( I_i = 0 \) if the \(i\)-th individual is on time.

This setup helps us calculate the number of people who are late as probability calculations on these variables.

The Cumulative Distribution Function (CDF) is a fundamental concept in probability theory, providing cumulative probabilities up to a certain point. It helps you understand the likelihood of a random variable being at or below a specific value.

In our example, the number of people who arrive late, denoted by \( X \), follows a specific distribution. The CDF for \( X \) gives us \( P(X \leq x) \), which means the probability of having up to \( x \) late arrivals. For each \( x \), you sum the probabilities of all occurrences \( X \leq x \).

The formula for the CDF, \( F(x) \), adds up probabilities obtained from the probability mass function (PMF). For example, if the probabilities are

• \( P(X = 0) = 0.07776 \)
• \( P(X = 2) = 0.2592 \)
• \( P(X = 4) = 0.20736 \)

Then, the CDF at \( X = 4 \) is given by \( F(4) = P(X \leq 4) = 0.07776 + 0.2592 + 0.20736 \). This helps compute probabilities over intervals, like \( P(2 \leq X \leq 6) \).

Probability theory is the branch of mathematics concerned with the analysis of random phenomena. The three couples and two individuals attending the seminar are a perfect illustration of how probability theory helps in real-world situations.

This field provides a framework for modeling uncertainty, using mathematical structure to define outcomes, probabilities, and the relationships between events. Assuming lateness of attendees is independent is a typical application of probability theory. Under this assumption, factors like different cars for each couple and individual affecting each other's timing are disregarded, focusing purely on individual probability calculations.

Key concepts of probability theory utilized in this scenario include:

• Independence: Each couple or individual arriving late is independent of the others.
• Probability Mass Function (PMF): Defines probabilities associated with each possible number of late arrivals, essential for calculating further probabilities and distributions.

This structured approach lets us predict late arrivals and plan accordingly, based on the statistical model developed.

Calculating probabilities involves combining mathematical rules to quantify the likelihood of different scenarios. Starting with known probabilities for individuals and couples being late, we build up to the probability mass function (PMF) for our total variable \( X \), representing the count of late arrivals.

The event that all attendees are on time is a multiplication of individual probabilities: \( (0.6)^5 \). For specific lateness scenarios like one couple being late, there's a need to account for all possible combinations, multiplying and summing them appropriately. Steps like these involve using combinations and formulas:

• Single Event Probability: \( P(X = 0)= (0.6)^5 = 0.07776 \)
• Combination-Based Probability: Different couple lateness scenarios such as two couples can be computed as \( \binom{3}{2}(0.4)^2(0.6)^3 \).

Using this approach leads to a detailed probability profile, covering all potential lateness scenarios. Subsequent calculations of cumulative probabilities from the CDF provide further insights, like \( P(2 \leq X \leq 6) \), to make informed judgments and predictions based on these probability models.