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A tree diagram is a helpful visual tool used in probability to outline all the possible outcomes of an experiment. In this case, we are interested in analyzing the probability of finding defective parts from a machine.

To construct a tree diagram, begin with a single node representing the start of our experiment. From this node, draw two branches representing the possible outcomes for the first trial: defective (D) and non-defective (ND). The probability of choosing a defective part is 0.03, and the probability of a non-defective part is 0.97.

Continue by branching out from the endpoints of the first branches for the second trial. Each branch from the first defective branch will again split into D and ND, with the same probabilities. Similarly, each branch from the first non-defective branch will also split into D and ND.

This tree diagram helps visually map out the pathways and calculate the probabilities of different scenarios, such as finding exactly one defective part. By assigning probabilities to each branch, you'll see all the potential outcomes and their likelihoods.

In probability calculations, identifying the likelihood of different outcomes is crucial. Using the tree diagram, we can easily calculate the probability of each scenario happening.

To find the probability of all outcomes where no defects are found, we look at the path where both parts are non-defective: (ND, ND). Calculate this using the formula:

• \( P(0) = 0.97 \times 0.97 = 0.9409 \)

This result shows there's a high chance of not finding any defect.

Next, to find the probability of exactly one defect, consider two paths: the first path (D, ND) and the second path (ND, D). Use the binomial formula to compute this probability:

• \( P(1) = \binom{2}{1} \times 0.03 \times 0.97 = 0.0582 \)

This indicates a moderate chance of finding exactly one defect.

For the probability of finding two defective parts (D, D), consider:

• \( P(2) = 0.03 \times 0.03 = 0.0009 \)

This is a very slim chance, reflecting the rarity of this outcome. Adding these probabilities should give us a total of 1, confirming the accuracy of the calculations.

Analyzing defective parts in a manufacturing process is essential for quality control. By understanding the likelihood of defects, manufacturers can improve processes and reduce waste.

In this exercise, the defective rate is given as 3%. Such information is vital to create strategies for reducing defects. Notably, this problem exemplifies a binomial experiment: evaluating two items where each can be defective or not. Each trial is independent, and the probability of defects is constant.

By examining the experimental outcomes, you can focus on pathways that lead to one defect. The outcomes (D, ND) and (ND, D) both result in exactly one defective part. With visual aids like tree diagrams, you can effortlessly identify these crucial pathways.

This method of analysis helps anticipate the proportion of defective parts over many trials, allowing fine-tuning of the machine or process for optimal efficiency. Keeping an eye on the defect probabilities enables companies to maintain high-quality production standards.