91Ó°ÊÓ

Understanding event probability is crucial when working with probability distributions. An event is any set of outcomes we are interested in, within a probability space. In our exercise, an event like "at most three lines are in use" involves specific outcomes from the random variable.

• "At most" means considering all values up to and including the designated number. For example, "at most three" means we consider x=0, x=1, x=2, and x=3.
• "Fewer than three" involves all values less than three, which would be x=0, x=1, and x=2.
• "At least three" means verifing all outcomes from three upwards, including x=3, x=4, x=5, and x=6.

Calculating the probability of these events involves summing the probabilities of these specified outcomes. For instance, using the data provided, the probability that at most three lines are in use is 0.70, by adding the probabilities of x-values 0 through 3. Calculating event probabilities helps to predict and understand potential outcomes in uncertain scenarios.

A random variable is a variable that takes on values determined by the outcome of a random phenomenon. In simple terms, it's how we represent outcomes in mathematical terms. In our exercise, the random variable is denoted by \(x\), representing the number of telephone lines in use.Random variables can be of different types: discrete random variables, which take on a countable number of distinct values, and continuous random variables, which can take on any value in a continuum.

• Our variable \(x\) is discrete, as it can only assume specific whole number values from 0 to 6, based on how many lines are in use.
• The actual value \(x\) takes depends on the outcome of the random process; here, how many customers are calling in at the same time.

By using random variables like \(x\), we're able to model and evaluate different scenarios involving uncertainty, like how many telephone lines are likely to be in use at any point in time. The accompanying probability distribution gives the likelihood of each possible number of lines being used.

Discrete probability is the branch of probability that deals with discrete random variables. Discrete random variables, like our \(x\) in the exercise, can only take specific, countable values. This makes calculating probabilities clearer, as we can list all possible outcomes.For each value that our random variable takes, a probability is assigned, which must meet these criteria:

• The probability of any specific outcome is between 0 and 1.
• The sum of all probabilities must equal 1, ensuring that they cover the entire sample space.
• For example, the probabilities for \(x=0,1,2,3,4,5,6\) add up to 1, confirming that all potential numbers of lines in use are accounted for.

This makes methods like summing probabilities straightforward for determining the likelihood of various events. Events are composed of multiple specific outcomes, and their probabilities reflect how likely each event is within the discrete framework. Discrete probability distributions thus serve as a powerful tool for predicting outcomes in environments with finite possibilities, aiding in decision-making and planning where exact numbers are involved.