/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 21 The birthday problem What鈥檚 th... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 21

The birthday problem What鈥檚 the probability that in a randomly selected group of 30 unrelated people, at least two have the same birthday? Let鈥檚 make two assumptions to simplify the problem. First, well ignore the possibility of a February 29 birthday. Second, we assume that a randomly chosen person is equally likely to be born on each of the remaining 365 days of the year. (a) How would you use random digits to imitate one repetition of the process? What variable would you measure? (b) Use technology to perform 5 repetitions. Record the outcome of each repetition. (c) Would you be surprised to learn that the theoretical probability is 0.71? Why or why not?

Short Answer

Expert verified
No, repetition results should align with a 0.71 probability, supporting it as reasonable.

Step by step solution

01

Set Up the Problem

We want to calculate the probability that at least two people in a group of 30 have the same birthday. To simplify, we assume there are 365 days in a year, and each day is equally likely for a birthday.
02

Define Random Digits for Simulation

To simulate this scenario, assign each of the 365 days (from 000 to 364) a unique 3-digit random number. Each person in the group will be assigned a random number representing a birthday.
03

Perform Simulation

For a simulation using random digits, generate 30 random numbers (birthdays) ranging from 000 to 364. Check if there are any duplicates among these 30 numbers to simulate one repetition of the process.
04

Measure the Variable of Interest

The variable to measure is whether at least two people have the same birthday in one simulation. Record 'Yes' if there is a duplicate and 'No' if there is not.
05

Repeat the Simulation

Use technology to perform 5 repetitions of this process. For each repetition, simulate 30 random birthdays and record if there is a match every time.
06

Analyze the Simulation Results

After running the simulations, count the number of times that at least one pair of people in the group of 30 shared the same birthday. Use this data to understand and compare against theoretical probability.
07

Compare with Theoretical Probability

The theoretical probability of at least two people sharing a birthday in a group of 30 is approximately 0.71. Based on your simulation (often resulting in a similar chance of matches), consider whether a probability of 0.71 seems surprising. Generally, seeing repetitions showcasing this probability evidence supports the theoretical probability.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Birthday Problem
The Birthday Problem is a fascinating probability puzzle that asks about the likelihood of at least two people sharing the same birthday in a group. This is unique because the probability becomes surprisingly high with relatively few people. For example, in a group of just 23 people, the chance of at least two sharing a birthday is over 50%. With 30 people, it jumps to about 71%. This counterintuitive result comes from the vast number of possible pairings between individuals, rather than the number of people itself. Understanding this helps us explore the principles of probability and combinatorics in everyday events.
Random Digits
Using random digits to simulate a scenario involves assigning numbers to represent different outcomes. In the birthday problem, each day of the year is assigned a number from 000 to 364. To simulate, generate 30 random numbers corresponding to birthdays. By doing this, you can imitate the distribution of birthdays in a group of people. Checking for duplicates among these numbers shows whether any pair shares a birthday. This technique is vital for essentially recreating real-world randomness in a controlled, repeatable way during simulations.
Theoretical Probability
Theoretical probability involves calculating the likelihood of an event based on assumed perfect conditions. For the birthday problem, it鈥檚 about predicting the chance that two or more people share a birthday in a group. To calculate, consider that the first person can have a birthday on any of the 365 days. The second person has a slightly reduced chance of not matching, and so on. The probability of no shared birthdays is calculated first, and is then subtracted from 1 to find the probability of at least one match. This approach gives a probability of about 0.71 for 30 people, illustrating just how theory can predict highly non-intuitive results.
Simulation Analysis
Simulation analysis helps to compare actual results with theoretical predictions. By repeating simulations, you get a clearer picture of how often an event like a birthday match occurs. In the birthday problem, by running the simulation multiple times 鈥� say, five repetitions 鈥� you can count how many times at least two people in the group share a birthday. The closer these results are to the theoretical probability, the more confident you can be in your understanding of probability theory in practice. It鈥檚 a powerful tool for validating theoretical models and demonstrating randomness.
Probability Theory
Probability theory is the mathematical framework for analyzing random phenomena. It focuses on predicting the must likely outcomes of uncertain events. Central to this theory is the concept of events, which are outcomes or happenings with a defined probability. In the birthday problem, the event is the occurrence of shared birthdays. Understanding probability theory helps unravel complex scenarios, illuminating the connections between seemingly random events and structured mathematical principles. It relies on concepts like sample space, events, and probability measures, providing the tools to analyze scenarios like the birthday problem accurately.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Treating low bone density (4.2) Fractures of the spine are common and serious among women with advanced osteoporosis (low mineral density in the bones). Can taking strontium ranelate help? A large medical trial assigned 1649 women to take either strontium ranelate or a placebo each day. All of the subjects had osteoporosis and had had at least one fracture. All were taking calcium supplements and receiving standard medical care. The response variables were measurements of bone density and counts of new fractures over three years. The subjects were treated at 10 medical centers in 10 different countries.9 Outline an appropriate design for this experiment. Explain why this is the proper design.

Late shows Some TV shows begin after their scheduled times when earlier programs run late. According to a network鈥檚 records, about 3% of its shows start late. To find the probability that three consecutive shows on this network start on time, can we multiply (0.97)(0.97)(0.97)? Why or why not?

At the gym Suppose that 10% of adults belong to health clubs, and 40% of these health club members go to the club at least twice a week. What percent of all adults go to a health club at least twice a week? Write the information given in terms of probabilities, and use the general multiplication rule.

Tall people and basketball players Select an adult at random. Define events T: person is over 6 feet tall, and B: person is a professional basketball player. Rank the following probabilities from smallest to largest. Justify your answer. $$ P(T) \quad P(B) \qquad P(T | B) \qquad P(B | T) $$

A probability teaser Suppose (as is roughly correct) that each child born is equally likely to be a boy or a girl and that the genders of successive children are independent. If we let BG mean that the older child is a boy and the younger child is a girl, then each of the combinations BB, BG, GB, and GG has probability 0.25. Ashley and Brianna each have two children. (a) You know that at least one of Ashley鈥檚 children is a boy. What is the conditional probability that she has two boys? (b) You know that Brianna鈥檚 older child is a boy. What is the conditional probability that she has two boys?
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Applied Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.