Question

More bad simulations. Explain why each of the following simulations fails to model the real situation: a) Use random numbers 2 through 12 to represent the sum of the faces when two dice are rolled. b) Use a random integer from 0 through 5 to represent the number of boys in a family of 5 children. c) Simulate a baseball player's performance at bat by letting \(0=\) an out, \(1=\) a single, \(2=\) a double, \(3=\) a triple, and \(4=\) a home run.

Short answer

All simulations fail due to incorrect probability assumptions, as they incorrectly assume uniform distributions.

Step-by-step solution

Analyzing Failure of Die Roll Simulation

When simulating the roll of two dice by generating random numbers from 2 through 12, this method fails because it assumes each sum from 2 to 12 is equally likely. However, the probabilities of these sums differ. For instance, 7 can be obtained in 6 different ways (e.g., 1+6, 2+5), making it the most common result, while 2 can only be obtained with (1+1). Thus, the probabilities are not uniform, and this simulation cannot accurately model two dice rolls.

Examining Family Composition Simulation

Using a random integer from 0 through 5 to represent the number of boys in a family with 5 children assumes that each outcome from 0 to 5 boys is equally likely. However, when modeling based on birth probabilities, we should assume each child is equally likely to be a boy or a girl, following a binomial distribution. This leads to varying probabilities for different numbers of boys, not a uniform probability.

Critiquing Player Performance Simulation

The simulation of a baseball player's performance at bat using integers 0 through 4 assumes all outcomes (outs, singles, doubles, triples, home runs) are equally likely. In reality, these events have different probabilities. Outs occur more frequently than home runs or triples. Hence, this equal-probability simulation fails to accurately represent the range of possible outcomes and their likelihoods.

Key concepts

Probability

Understanding probability is essential for modeling real-world scenarios accurately. Probability is a measure of how likely it is for a particular event to occur within a set of possible outcomes. For instance, when rolling two dice, each die has 6 faces, leading to 36 total outcomes (since 6 x 6 = 36). However, not all possible sums from rolling two dice are equally probable. Here are some key points to remember:

• The sum of 7 is the most likely because there are multiple combinations (like (1+6), (2+5), etc.) that result in this sum.
• The sum of 2 or 12 is less likely, with only one combination for each (1+1 for 2, and 6+6 for 12).

When simulating events, it is crucial to consider the probabilities of different outcomes, so they reflect real-world possibilities rather than assuming all outcomes are equally likely.

Binomial Distribution

The binomial distribution is a fundamental concept when it comes to modeling scenarios where there are two possible outcomes, like determining the number of boys or girls in a family of five children. This distribution captures the probabilities of obtaining different counts of successes (e.g., boys) in several independent trials (e.g., having five children).In a binomial distribution:

• Each trial has the same probability of success, denoted by \( p \).
• The total number of trials is represented by \( n \).
• The probability of observing \( k \) successes, here the number of boys, is calculated using the formula: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]

Applying this to our family example, if the probability of having a boy is 0.5 (assuming boys and girls are equally likely), the distribution would give different probabilities for 0 to 5 boys, rather than assuming each count is equally probable.

Dice Rolling

Dice rolling is a classic problem in probability, often used to introduce randomness and probability distribution concepts. When rolling two dice, we commonly deal with sums ranging from 2 to 12. However, as previously mentioned, these sums are not equally likely. To better understand this:

• With two dice, there are 36 possible outcomes, considering each die is independent.
• The outcome of 7 is common due to the highest number of combinations (such as (1+6), (2+5), etc.) resulting in that sum.
• A sum like 2 or 12 occurs with the lowest frequency, each obtainable with only one combination.

Accurate simulations should take this unequal distribution into account, reflecting how dice actually behave based on their physical properties and probability theory.

Random Numbers

Random numbers form the backbone of simulations, allowing us to mimic randomness found in nature and various applications like gaming or statistical modeling. However, using them correctly is vital to create realistic simulations. Points to consider in simulations:

• Random numbers help model chance events but should match the probability distribution of real-world scenarios.
• For instance, assigning random numbers from 2 to 12 to dice rolls fails, as discussed, because it suggests an incorrect probability distribution of sums.
• Similarly, for a family size simulation or sports performance modeling, the base probabilities for outcomes need to match what's observed in reality.

Thus, choosing the correct range and matching probabilities to your simulation's real-world counterpart is essential for accuracy and reliability.