Is this valid? Determine whether each of the following...

Is this valid? Determine whether each of the following simulation designs is valid. Justify your answer. (a) According to a recent survey, $50 \%$ of people aged 13 and older in the United States are addicted to texting. To simulate choosing a random sample of 20 people in this population and seeing how many of them are addicted to texting, use a deck of cards. Shuffle the deck well, and then draw one card at a time. A red card means that person is addicted to texting; a black card means he isn't. Continue until you have drawn 20 cards (without replacement) for the sample. (b) A tennis player gets $95 \%$ of his serves in play during practice (that is, the ball doesn't go out of bounds). To simulate the player hitting 5 serves, look at 5 pairs of digits going across a row in Table $D .$ If the number is between 00 and 94 , the serve is in; numbers between 95 and 99 indicate that the serve is out.

Is this valid? Determine whether each of the following simulation designs is valid. Justify your answer. (a) According to a recent survey, $50 \%$ of people aged 13 and older in the United States are addicted to texting. To simulate choosing a random sample of 20 people in this population and seeing how many of them are addicted to texting, use a deck of cards. Shuffle the deck well, and then draw one card at a time. A red card means that person is addicted to texting; a black card means he isn't. Continue until you have drawn 20 cards (without replacement) for the sample.
(b) A tennis player gets $95 \%$ of his serves in play during practice (that is, the ball doesn't go out of bounds). To simulate the player hitting 5 serves, look at 5 pairs of digits going across a row in Table $D .$ If the number is between 00 and 94 , the serve is in; numbers between 95 and 99 indicate that the serve is out.

Key Concepts

Probability Models (Binomial vs. Hypergeometric Distribution)

Different probability models describe different types of sampling processes. The binomial model applies to situations with fixed probabilities in independent trials (typically involving sampling with replacement or sampling from an infinite population), whereas the hypergeometric model applies to sampling without replacement from a finite population. Recognizing which model best corresponds to the real-life scenario being simulated is essential to determine if the simulation method produces valid and accurate results.

Randomization Techniques

Randomization techniques involve generating outcomes in a manner that is free from systematic bias, ensuring that the simulation accurately reflects randomness inherent in the underlying process. The quality of a simulation hinges on the device or procedure used—whether it is a physical medium like a deck of cards or a digital table of random numbers—to guarantee that each potential outcome occurs with the proper relative frequency dictated by the theoretical model.

Sampling Methods

This concept refers to the procedures used to select individuals or outcomes for the simulation. In simulation design, maintaining the intended probabilities often requires careful consideration of whether sampling is performed with or without replacement. Sampling without replacement can lead to dependent outcomes, which may deviate from the intended independent probability model, whereas sampling with replacement helps ensure that each selection is independent and has the same probability distribution as the overall population.

Independence of Trials

Independence of trials is a crucial concept in probability and simulation. Each event or trial should occur without being affected by the outcomes of previous events, ensuring that the probability remains constant across trials. In simulation designs, a method that inadvertently introduces dependence (for example, due to a diminishing pool of outcomes) can invalidate the generated probability model if the goal is to mimic independent events such as those found in a binomial setting.

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