Determine whether each of the fallowing simulation designs...

Determine whether each of the fallowing simulation designs is valid. Justify your answer. (a) According to a recent survey, $50 \%$ of people. aped 13 and alder in the United States are addicted to email. To simulate choosing a random sample of 20 people in this population and seeing how many of them are addicted to email, use u deck of cards. Shuffle the deck well, and then draw one card at a time. $A$ red card means that person is addicted to email; a black card means he isn't. Continue until you have drawn 20 cards i without replacement for the sample. (b) $A$ tennis player gets $95 \%$ of his second serves in play during practice that is, the ball doesn't go out of bounds). To simulate the player hitting, 5 second serves, look at pairs of digits going across in row in Table $D$. If the number is between 00 and $94,$ the serve is in numbers between 95 and $\$ 9$ indicate that the serve is out.

Determine whether each of the fallowing simulation designs is valid. Justify your answer.
(a) According to a recent survey, $50 \%$ of people. aped 13 and alder in the United States are addicted to email. To simulate choosing a random sample of 20 people in this population and seeing how many of them are addicted to email, use u deck of cards. Shuffle the deck well, and then draw one card at a time. $A$ red card means that person is addicted to email; a black card means he isn't. Continue until you have drawn 20 cards i without replacement for the sample.
(b) $A$ tennis player gets $95 \%$ of his second serves in play during practice that is, the ball doesn't go out of bounds). To simulate the player hitting, 5 second serves, look at pairs of digits going across in row in Table $D$. If the number is between 00 and $94,$ the serve is in numbers between 95 and $\$ 9$ indicate that the serve is out.

Key Concepts

Sampling with or without replacement

In simulation experiments, it’s critical to mimic the theoretical conditions of the process being modeled. One common issue is whether sampling is done with replacement (ensuring each trial is independent and the probability remains constant) or without replacement (where the outcome of one draw affects the probabilities of the subsequent draws). A simulation design that uses methods not matching the originally stated probabilities can introduce bias and lead to incorrect results.

Independence of Simulation Trials

Many simulations assume that each trial is independent, meaning that the outcome of one trial does not affect the others. This is crucial, especially when simulating events that are meant to have identical likelihoods across multiple trials. Any method that inadvertently introduces dependence between outcomes might misrepresent the model’s random behavior, leading to an invalid simulation.

Mapping Numerical Outcomes to Probabilities

A key aspect of designing a valid simulation is the correct matching of the random mechanism’s outcomes to the event probabilities. This involves partitioning the set of possible outcomes (like numbers or symbols) into groups that accurately reflect the theoretical probability model. If the mapping is incorrect or if the random device does not generate outcomes uniformly over the desired range, the simulation will not accurately reflect the intended probability distribution.

Use of Random Digit Tables

Random digit tables (or other sources of random numbers) are often used in simulations because they are assumed to provide uniform randomness. When these tables are used appropriately—by correctly grouping outcomes to match event probabilities—they yield valid simulation results. The validity depends on the assumption that each group of digits corresponds exactly to the desired probability events without introducing bias.

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