Question

Bad simulations. Explain why each of the following simulations fails to model the real situation properly: a) Use a random integer from 0 through 9 to represent the number of heads when 9 coins are tossed. b) A basketball player takes a foul shot. Look at a random digit, using an odd digit to represent a good shot and an even digit to represent a miss. c) Use random digits from 1 through 13 to represent the denominations of the cards in a five-card poker hand.

Short answer

Each simulation method oversimplifies the situation, misrepresenting probabilities or distributions.

Step-by-step solution

Understanding Coin Toss Probabilities

In a fair 9-coin toss, each coin has a 50% chance of landing heads or tails. Therefore, the number of heads is represented by a binomial distribution, which doesn't match the uniform distribution generated by a random integer from 0 to 9. This method doesn't account for the different probabilities of getting specific numbers of heads, such as 4 heads being more likely than 0 or 9 heads.

Analyzing Basketball Shot Representation

A basketball player's shot success rate isn't exactly 50/50, so using an odd digit for a basket and an even digit for a miss assumes this probability, which likely doesn't match the player's actual success rate. Also, this method suggests perfect independence and equal chances, which may not capture streaks or changing probabilities during gameplay.

Rationalizing Card Denomination Simulation

In a standard deck, denominations from Ace through King are mapped to numbers 1 through 13. However, this approach assumes equal likelihood for each number to appear, ignoring the different card suits. The method also does not represent the actual composition of a poker hand, which should be a random selection from a finite set of cards.

Key concepts

Probability

Probability is the fundamental concept used to measure the likelihood of an event occurring. Imagine tossing a single coin—it has two possible outcomes: heads or tails. Each outcome has a probability of 0.5 or 50%, assuming a fair coin.
In more complex scenarios, like the exercises mentioned, probability helps to determine the likely outcomes that align with real-world situations. It is important to match the probability of an event with the appropriate model to ensure simulations are accurate and reflective of true scenarios.

• In coin-tossing, probabilities should reflect the chance of multiple heads, which is better captured by a distribution rather than just a single random number.
• For basketball shots, real-life shot probabilities are crucial to account for skills and conditions, as opposed to a simple odd-even approach.

Understanding probability ensures our simulations don't misrepresent the true likelihood of events.

Binomial Distribution

The binomial distribution is a specific probability distribution that applies to situations with two possible outcomes (like heads or tails in coin flips) done over a set number of trials. For example, when tossing 9 coins, the result can be perfectly described by a binomial distribution because each coin toss is an independent trial with a success (heads) or failure (tails).
This distribution allows us to calculate the probability of achieving a specific number of successes across multiple trials. For instance, getting exactly four heads when tossing 9 coins can be predicted using the formula for binomial probability:$$P(X=k)=(\genfrac{}{}{0ex}{}{n}{k})p^k(1-p{)}^{n-k}$$where:

• $n$ is the number of trials (9 coins)
• $k$ is the desired number of successes (e.g., 4 heads)
• $p$ is the probability of success on a single trial (0.5 for heads)

Using this model is essential to accurately simulate scenarios where events have two outcomes repeated several times.

Randomness

Randomness refers to the lack of pattern or predictability in events. It is a crucial element when trying to simulate real-world processes. In sports and games like poker, randomness ensures that each event is independent of previous events.
For example, in the simulation of a basketball player's shots, if we assume each shot is determined by randomness but ignore actual probability, we fail to capture the essence of the player's performance. True randomness in simulations should mimic the unpredictable nature of the real-world, yet still account for known probabilities of outcomes.
An ideal simulation needs to blend randomness with reality-based probability, ensuring each event's chance is as realistic as possible, reflecting both the random and pattern elements found in the real world.

Uniform Distribution

A uniform distribution is one where every outcome is equally likely. Think of rolling a fair six-sided die—all outcomes from 1 to 6 have the same chance of occurring. For simulations, this distribution fits scenarios where each potential outcome truly has an equal probability.
However, the misuse of uniform distribution is common. In the simulation exercises given, it was improperly used to simulate scenarios where outcomes do not actually have equal chances. For instance, in selecting cards from a deck, simply using numbers 1 through 13 doesn’t account for the different suits or the constraint of drawing from a finite deck.
Since real-world scenarios like these have different probabilities, to simulate them accurately, you need a model that reflects those varying chances instead of assuming uniformity.