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The term "sample space" refers to all possible outcomes of a particular random experiment. In the exercise about inspecting switches, the sample space includes all possible sequences of outcomes when the inspector draws two switches in succession.
It consists of four potential scenarios:

• Drawing two good switches.
• Drawing a good switch and then a bad switch.
• Drawing a bad switch and then a good switch.
• Drawing two bad switches.

Understanding the sample space helps in visualizing all potential outcomes and is the foundation for evaluating probabilities in different scenarios. This structure allows us to organize and study approach probabilities effectively.

A tree diagram is a graphical representation used to map out the sample space of a chance experiment step by step. It is a helpful tool for visualizing the process of random events and systematic probabilities.
In this problem, the tree diagram begins with two initial branches. Each branch stems from drawing either a good switch or a bad switch during the first draw. Each of these branches then splits again based on the outcome of the second draw. If the first switch drawn is good, the second set of branches represents whether the next switch is good or bad.
Tree diagrams are especially useful in illustrating the outcomes of the sequence and showing how probabilities change with each event. It clearly arranges and depicts possible outcomes, making it easy to calculate probabilities by following the branches.

Dependent events are those where the outcome or occurrence of the first event affect the outcome or occurrence of the second event.
In the context of the switch inspection, the events are dependent because the result of the first draw influences the probabilities in the second draw. For example:

• When a defective switch is drawn first, there is one fewer bad switch in the pool, affecting the subsequent probabilities.
• This affects the probability calculations because the total number of switches and the number of defective switches change with each draw.

Understanding dependent events is crucial as it reveals that prior outcomes influence subsequent probabilities. This understanding ensures more accurate calculation of probabilities and enhances the depth of problem-solving skills.

Calculating probability involves using the outcomes from the sample space and the structured paths depicted in the tree diagram. In the inspection problem, probability calculation is about determining the likelihood of certain events, such as drawing two defective switches.
To calculate the probability of both switches being defective:

• First, determine the probability of drawing a bad switch in the first draw: \( \frac{1000}{10000} \).
• Then, calculate the probability of the second switch being bad given that the first one was bad: \( \frac{999}{9999} \).
• The total probability is the product of these: \( \frac{1000}{10000} \times \frac{999}{9999} \).

Multiply these probabilities to get \( \approx 0.00999 \), indicating a very small chance of randomly selecting two bad switches consecutively. Such calculations show how probabilities can be derived using structured logic and are crucial for analyzing real-life processes involving chance.