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A probability distribution table is an organized way to display the possible outcomes of a random variable and their corresponding probabilities. For the given problem, we are looking at two employees and determining how many of them are happy with their jobs. Here, the random variable \(x\) represents the number of happy employees out of two.

We calculate the probabilities for each possible value of \(x\) using the binomial distribution formula. This formula helps us find the likelihood of obtaining a fixed number of successes (happy employees) in a certain number of trials (the two selected employees).

The probabilities need to add up to 1, meaning all possible outcomes are accounted for. In our table:

• \(x = 0\): The probability that neither employee is happy is \(0.4225\).
• \(x = 1\): The probability that exactly one employee is happy is \(0.455\).
• \(x = 2\): The probability that both employees are happy is \(0.1225\).

This table helps us understand the possible scenarios and their likelihood, providing a clear overview of the distribution of happiness among the employees.

A tree diagram is a visual representation that outlines all the possible outcomes of an event and their probabilities. It is particularly useful for illustrating a sequence of possible events, like in our exercise where the happiness states of two employees are considered separately.

In this problem, the tree diagram consists of two stages: Firstly, we look at the outcome of the first employee being either Happy (H) or Not Happy (NH). Secondly, we analyze the outcome of the second employee, given the outcome of the first.

The diagram splits into branches at each stage:

• Stage 1 (First Employee): Two branches - \(0.35\) for H and \(0.65\) for NH.
• Stage 2 (Second Employee):
  • If the first employee is NH, two further branches: NH-NH and NH-H with probabilities \(0.65\) and \(0.35\) respectively.
  • If the first employee is H, two further branches: H-NH and H-H with probabilities \(0.65\) and \(0.35\) respectively.

By using the tree diagram, we can visually trace the paths and calculate the probability of each final outcome by multiplying the probabilities along the branches.

Calculating probabilities in a binomial distribution helps us find out the individual likelihood of various outcomes within the experiment. In this case, the binomial distribution formula \(P(X=k)=\binom{n}{k} p^{k} (1-p)^{n-k}\) is employed to derive the probabilities for \(x = 0\), \(x = 1\), and \(x = 2\).

Let's break this formula down for a clearer understanding:

• \(n\) is the number of trials, which is 2 (since we are looking at two employees).
• \(k\) is the number of successful outcomes (happy employees). We calculate it for \(k = 0, 1, 2\).
• \(p\) is the probability of success (probability that an employee is happy) which is \(0.35\).
• \(\binom{n}{k}\) is a combination function to determine the number of ways \(k\) happy employees can occur out of \(n\).

Applying this formula:

• \(x = 0\): None are happy, resulting in a probability of \(0.4225\).
• \(x = 1\): Exactly one is happy, with a calculated probability of \(0.455\).
• \(x = 2\): Both are happy, resulting in a probability of \(0.1225\).

This systematic approach ensures all possible scenarios are evaluated accurately, helping to understand the overall happiness levels within the organization.