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Chapter 4: Problem 18

a. Develop a simulation for finding the probability that when 50 people are randomly selected, at least 2 of them have the same birth date. Describe the simulation and estimate the probability. b. Develop a simulation for finding the probability that when 50 people are randomly selected, at least 3 of them have the same birth date. Describe the simulation and estimate the probability.

Short Answer

Expert verified
To estimate the probability, simulate birthdays of 50 people multiple times. Check for shared birthdays. Count and divide by the number of trials for percent estimates.

Step by step solution

01

- Define the Problem

We need to estimate the probability that among 50 randomly chosen people, at least 2 (for part a) and at least 3 (for part b) have the same birthday.
02

- Set Up the Simulation

To simulate this, consider a year with 365 days. Each person can be assigned a random birthday from these 365 days. This process needs to be repeated for 50 people.
03

- Simulate Multiple Trials for Part a

Create a large number of trials where you simulate the birthdays of 50 people. In each trial, check if at least 2 people share a birthday. Count how many trials result in at least one shared birthday.
04

- Calculate the Probability for Part a

The probability of at least 2 people sharing a birthday is the number of successful trials divided by the total number of trials.
05

- Simulate Multiple Trials for Part b

Similarly, create a large number of trials for 50 people. In each trial, check if at least 3 people share the same birthday. Count how many trials result in this condition being met.
06

- Calculate the Probability for Part b

The probability of at least 3 people sharing a birthday is the number of successful trials divided by the total number of trials.
07

- Estimate the Probabilities

Estimate the probabilities based on the results from the simulations. The more trials performed, the more accurate the probability estimates will be.

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Key Concepts

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

Simulation
Simulations are powerful tools in probability and statistics. They help us visualize and understand complex problems through repeated experiments. In the context of the birthday problem, a simulation involves creating a model of a scenario where we randomly assign birthdays to a group of people. By doing this multiple times, we can observe patterns and estimate probabilities.
Probability Estimation
To estimate the probability of at least two people sharing a birthday in a group of 50, we rely on the results of our simulation. Here's how:
  • Run numerous trials of our model, where we randomly assign birthdays to 50 people.
  • In each trial, check if at least two people share a birthday.
  • Count how many of these trials meet this condition.
  • Divide the number of successful trials by the total number of trials to find the estimated probability.
The more trials you run, the more accurate your probability estimate will be.
Random Selection
Random selection is crucial for the birthday problem simulations. Each individual's birthday is chosen randomly from one of 365 days of the year. This ensures that each person has an equal chance of being born on any given day. By mimicking real-life randomness in our simulations, we can gain reliable insights into the likelihood of shared birthdays.
Repeated Trials
Repeated trials form the backbone of a simulation's accuracy. Instead of relying on a single run, we repeat the simulation many times. This repetition helps account for the variability inherent in random events.
  • For Part a, repeat the process of assigning birthdays to 50 people and checking for shared birthdays many times.
  • For Part b, similarly repeat the simulation, but now look for at least three people sharing a birthday.
More trials lead to more precise probability estimates.
Birthdays
Understanding the concept of birthdays within the context of this problem is key. Here, a birthday is represented as a day of the year (ignoring leap years for simplicity).

In our simulation, we're interested in the phenomenon where two or more people out of 50 share the same birth date. By using this birthday simulation model, we tackle a classic probability puzzle that initially might seem counterintuitive but reveals intriguing insights upon closer examination.

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Most popular questions from this chapter

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.) Shaquille O'Neal was a professional basketball star who had a reputation for being a poor free-throw shooter. In his career, he made 5935 of 11,252 free throws that he attempted, for a success ratio of \(0.527\). Describe a procedure for using software or a TI- \(83 / 84\) Plus calculator to simulate his next free throw. The outcome should be an indication of one of two results: (1) The free throw is made; (2) the free throw is missed.

Find the indicated complements. A U.S. Cellular survey of smartphone users showed that \(26 \%\) of respondents answered "yes" when asked if abbreviations (such as LOL) are annoying when texting. What is the probability of randomly selecting a smartphone user and getting a response other than "yes"?

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.) When Mendel conducted his famous hybridization experiments, he used peas with green pods and yellow pods. One experiment involved crossing peas in such a way that \(75 \%\) of the offspring peas were expected to have green pods, and \(25 \%\) of the offspring peas were expected to have yellow pods. Describe a procedure for using software or a TI-83/84 Plus calculator to simulate 20 peas in such a hybridization experiment. Each of the 20 individual outcomes should be an indication of one of two results: (1) The pod is green; (2) the pod is yellow.

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.) The probability of randomly selecting an adult who recognizes the brand name of McDonald's is \(0.95\) (based on data from Franchise Advantage). Describe a procedure for using software or a TI-83/84 Plus calculator to simulate the random selection of 50 adult consumers. Each individual outcome should be an indication of one of two results: (1) The consumer recognizes the brand name of McDonald's; (2) the consumer does not recognize the brand name of McDonald's.

Express all probabilities as fractions. With a short time remaining in the day, a FedEx driver has time to make deliveries at 6 locations among the 9 locations remaining. How many different routes are possible?
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