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A probability distribution is a fundamental concept in probability theory that describes the likelihood of different outcomes for a random variable. In the context of our exercise, the random variable is the dental insurance payout, denoted as \(X\). This random variable can take on specific values depending on the occurrence of certain events over the next three years.

Here’s how it works:

• The probability distribution assigns a probability to each of these outcomes, reflecting how likely each event is to happen.
• For our example, the events are the need for a major repair, a minor repair, or no repair, each with assigned probabilities.

This means:

• The probability \(P(X = 1000)\) of receiving a \\(1000 payout (major repair) is \(5\%\) or 0.05.
• For a \\)100 payout (minor repair), \(P(X = 100)\) is \(60\%\) or 0.60.
• Lastly, the probability \(P(X = 0)\) for no payout (no repair) is \(35\%\) or 0.35.

This approach allows us to fully understand and anticipate the chances of different payouts from the dental insurance policy.

A discrete random variable is a type of random variable that has a countable number of possible values. Unlike continuous random variables that can take any value in a range, discrete random variables are limited to specific outcomes. In our insurance example, \(X\) is a discrete random variable because it only takes certain specific values based on the dental repair events.

Consider the following:

• The payouts provided by the insurance—\\(1000, \\)100, and \$0—are distinct, separate numbers.
• These values are not part of a continuous range of numbers; they are isolated points.
• This defines \(X\) as discrete, highlighting scenarios where each outcome is distinct and intrinsic to the event.

Thus, understanding \(X\) as a discrete random variable helps students predict payment outcomes accurately based on their categorical nature.

Probability theory is the branch of mathematics that deals with analyzing and understanding random events. It provides the mathematical foundation for working with random variables and distributions, helping us assess uncertainty and chance.

In our exercise, we used probability theory to model and calculate the chance of different payout outcomes for the dental insurance. Applying probability theory involves:

• Identifying all potential outcomes (major repair, minor repair, no repair).
• Assessing their likelihood using probability rules.
• Summing probabilities to ensure they total 1, covering all possible scenarios.

The theory is critical in making predictions about future events based on historical data, helping insurance companies and policyholders assess risk and make informed decisions. In this case, through probability theory, we gain a clearer picture of expected payouts and their associated risks.