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Chapter 11: Problem 12

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

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

Step by step solution

01

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.
02

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.
03

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Birth defects. The American College of Obstetricians and Gynecologists says that out of every 100 babies born in the United States. 3 have some kind of major birth defect. How would you assign random numbers to conduct a simulation based on this statistic?

A bigger family. Suppose a couple will continue having children until they have at least two children of each sex (two boys and two girls). How many children might they expect to have?

The World Series. The World Series ends when a team wins 4 games. Suppose that sports analysts consider one team a bit stronger, with a \(55 \%\) chance to win any individual game. Estimate the likelihood that the underdog wins the series.

Beat the lottery. Many states run lotteries to raise money. A Web site advertises that it knows "how to increase YOUR chances of Winning the Lottery." They offer several systems and criticize others as foolish. One system is called Lucky Numbers. People who play the Lucky Numbers system just pick a "lucky" number to play, but maybe some numbers are luckier than others. Let's use a simulation to see how well this system works. To make the situation manageable, simulate a simple lottery in which a single digit from 0 to 9 is selected as the winning number. Pick a single value to bet, such as 1 , and keep playing it over and over. You'll want to run at least 100 trials. (If you can program the simulations on a computer, run several hundred. Or generalize the questions to a lottery that chooses two- or three-digit numbers-for which you'll need thousands of trials.) a) What proportion of the time do you expect to win? b) Would you expect better results if you picked a "luckier" number, such as \(7 ?\) (Try it if you don't know.) Explain.

Blood donors. A person with type O-positive blood can receive blood only from other type \(\mathrm{O}\) donors. About \(44 \%\) of the U.S. population has type \(\mathrm{O}\) blood. At a blood drive, how many potential donors do you expect to examine in order to get three units of type \(\mathrm{O}\) blood?
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