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Chapter 4: Problem 6

Describe the simulation procedure. (For example, to simulate 10 births, use a random number generator to generate 10 integers between 0 and 1 inclusive, and consider 0 to be a male and 1 to be a female.) Lefties Ten percent of people are left-handed. In a study of dexterity, 15 people are randomly selected. Describe a procedure for using software or a TI- \(83 / 84\) Plus calculator to simulate the random selection of 15 people. Each of the 15 outcomes should be an indication of one of two results: (1) Subject is left-handed; (2) subject is not left-handed.

Short Answer

Expert verified
Use a random number generator to generate 15 numbers between 0 and 1. If a number is less than 0.1, the person is left-handed; otherwise, not.

Step by step solution

01

Understand the Problem

We need to simulate the random selection of 15 people, with each outcome indicating if the subject is left-handed or not. Since 10% are left-handed, we'll use a random number generation method to decide for each individual.
02

Define the Probability

Since 10% of people are left-handed, the probability, denoted by P(LH), that any given person is left-handed is 0.1 (10%). Consequently, the probability that any given person is not left-handed is 0.9 (90%).
03

Set Up the Random Number Generator

Use a random number generator to produce random numbers between 0 and 1 for each person. Numbers in the interval [0, 0.1) will indicate a left-handed person (LH), and numbers in the interval [0.1, 1] will indicate a right-handed person (not LH).
04

Simulation Procedure in Software

Using software like Python, R, or any statistical software: - Generate 15 random numbers between 0 and 1. - For each number, determine if it's less than 0.1. - If it is, mark the person as left-handed (LH); otherwise, mark them as not left-handed (not LH).
05

Simulation Procedure Using TI-83/84 Plus Calculator

On the TI-83/84 Plus calculator: - Use the `rand` function to generate random numbers between 0 and 1. - Press `MATH`, arrow over to `PRB`, and select `1:rand`. - Repeat this process 15 times. - For each generated number, determine if it's less than 0.1. If so, mark the person as left-handed (LH); otherwise, mark them as not left-handed (not LH).
06

Record the Outcomes

Document the simulation results for the 15 people, indicating whether each person is left-handed or not according to the random number generated.

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Key Concepts

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

Probability Simulation
Probability simulation is a method for modeling real-world scenarios using random numbers. It involves generating random values to study phenomena that might be difficult to observe directly. For example, if we want to estimate how many people in a group might be left-handed, we can simulate this using probability principles. By knowing that 10% of people are left-handed, we use this probability in our simulations to make predictions on outcomes. It鈥檚 a powerful tool in statistics for understanding patterns and probabilities.
Random Number Generation
Random number generation is crucial in simulations. It involves creating a sequence of numbers that lack any predictable pattern. In our exercise, we used a random number generator to create numbers between 0 and 1. If a number falls in a specific range, we classify it accordingly. For instance, for our left-handed simulation, if a generated number is less than 0.1, it represents a left-handed person. This method ensures fairness and randomness in selecting our simulation subjects.
There are many tools available for random number generation, from simple functions in calculators to sophisticated algorithms in software like Python and R.
Left-Handed Probability
The concept of left-handed probability involves understanding that 10% of people are left-handed. This probability is denoted as P(LH) = 0.1. Conversely, the probability that someone is not left-handed, P(not LH), is 0.9 or 90%. When simulating a selection of people, we use these probabilities to determine outcomes. For each person in our simulation, there's a 10% chance they will be left-handed. This understanding helps us set the criteria for our random number generation, ensuring that our simulations align with real-world probabilities.
TI-83/84 Plus Calculator
The TI-83/84 Plus calculators are versatile tools capable of performing various statistical simulations. For our left-handed simulation, we utilize the `rand` function. Here's how you can do it:
  • Press the `MATH` button.
  • Navigate to the `PRB` (probability) menu.
  • Select `1:rand` to generate random numbers between 0 and 1.
  • Repeat this 15 times for our 15 subjects.
For each random number, if it鈥檚 less than 0.1, the person is marked as left-handed. This process helps perform our simulation efficiently and accurately, mimicking the probability setup we defined earlier.
Statistical Software
Statistical software, like Python, R, and others, offers powerful tools for simulations. By using built-in functions for random number generation, we can simulate complex scenarios effortlessly. For our left-handed probability exercise:
  • We write code to generate 15 random numbers between 0 and 1.
  • We then check each number: if less than 0.1, it indicates a left-handed individual.
  • Otherwise, it represents a right-handed person.
Software provides accurate, repeatable results and allows for more extensive simulations beyond the capabilities of simple calculators. It鈥檚 essential to document each output which helps in analyzing results and verifying our probability estimates.

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Most popular questions from this chapter

a. Five 鈥渕athletes鈥� celebrate after solving a particularly challenging problem during competition. If each mathlete high fives each other mathlete exactly once, what is the total number of high fives? b. If n mathletes shake hands with each other exactly once, what is the total number of handshakes? c. How many different ways can five mathletes be seated at a round table? (Assume that if everyone moves to the right, the seating arrangement is the same.) d. How many different ways can n mathletes be seated at a round table?

Redundancy in Computer Hard Drives Assume that there is a \(3 \%\) rate of disk drive failures in a year (based on data from various sources including lifehacker.com). a. If all of your computer data is stored on a hard disk drive with a copy stored on a second hard disk drive, what is the probability that during a year, you can avoid catastrophe with at least one working drive? Express the result with four decimal places. b. If copies of all of your computer data are stored on three independent hard disk drives, what is the probability that during a year, you can avoid catastrophe with at least one working drive? Express the result with six decimal places. What is wrong with using the usual round-off rule for probabilities in this case?

In the New Jersey Pick 6 lottery game, a bettor selects six different numbers, each between 1 and 49. Winning the top prize requires that the selected numbers match those that are drawn, but the order does not matter. Do calculations for winning this lottery involve permutations or combinations? Why?

Probability of At Least One Let \(A=\) the event of getting at least 1 defective iPhone when 3 iPhones are randomly selected with replacement from a batch. If \(5 \%\) of the iPhones in a batch are defective and the other \(95 \%\) are all good, which of the following are correct? a. \(P(\bar{A})=(0.95)(0.95)(0.95)=0.857\) b. \(P(A)=1-(0.95)(0.95)(0.95)=0.143\) c. \(P(A)=(0.05)(0.05)(0.05)=0.000125\)

Express all probabilities as fractions. You want to obtain cash by using an ATM, but it's dark and you can't see your card when you insert it. The card must be inserted with the front side up and the printing configured so that the beginning of your name enters first. a. What is the probability of selecting a random position and inserting the card with the result that the card is inserted correctly? b. What is the probability of randomly selecting the card's position and finding that it is incorrectly inserted on the first attempt, but it is correctly inserted on the second attempt? (Assume that the same position used for the first attempt could also be used for the second attempt.) c. How many random selections are required to be absolutely sure that the card works because it is inserted correctly?
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