Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 5: Problem 20

The Birthday Problem. The birthday problem is: if there is a group of \(n\) people in a room, what is the probability that two or more of them have the same birthday? It is possible to determine the answer to this question by simulation. Write a function that calculates the probabulity that two or more of \(n\) people will have the same birthday, where \(n\) is a calling argument. (Hint: To do this, the function should create an array of size \(n\) and generate \(n\) birthdays in the range 1 to 365 randomly. It should then check to see if any of the \(n\) birthdays are identical. The function should perform this experiment at least 5000 times and calculate the fraction of those times in which two or more people had the same birthday.) Write a test program that calculates and prints out the probability that 2 or more of \(n\) people will have the same birthday for \(n=2,3, \ldots, 40\).

Short Answer

Expert verified
The probability that 2 or more people out of \(n\) have the same birthday can be calculated by creating a function that generates \(n\) random birthdays, checks for any duplicates, performs this experiment 5000 times, and calculates the number of times duplicates were found divided by the number of experiments. This process is repeated for \(n\) ranging from 2 to 40. The Python implementation includes a function, calculate_birthday_probability, to perform these operations and a loop to execute this function for the required range of \(n\).

Step by step solution

01

1. Generate random birthdays

First, create a function that generates n random birthdays. To generate a random birthday, use a random number generator that returns a number in the range of 1 to 365. Create an array of size n to store these random birthdays for all n people.
02

2. Check for duplicate birthdays

In the same function, implement a method to check for any duplicate birthdays. You can use a set to store unique birthdays, and if the set's size is smaller than the array size, there is at least one duplicate birthday.
03

3. Perform the simulation and calculate the probability

Perform this experiment at least 5000 times in the function, and count how many times two or more people have the same birthday. To calculate the probability, divide the number of times with identical birthdays by the total number of experiments (5000).
04

4. Test the function for n = 2, 3, ..., 40

Execute the function for n = 2, 3, ..., 40, and print the probability in each case. To accomplish this, use a loop to call the function for each value of n in the specified range. Here is an implementation in Python: ```python import random def calculate_birthday_probability(n): duplicate_count = 0 for _ in range(5000): birthdays = [random.randint(1, 365) for _ in range(n)] unique_birthdays = set(birthdays) if len(unique_birthdays) < len(birthdays): duplicate_count += 1 probability = duplicate_count / 5000 return probability for n in range(2, 41): probability = calculate_birthday_probability(n) print(f"For n = {n}, the probability is: {probability}") ``` This implementation follows the steps as described and calculates the probability that 2 or more people have the same birthday for n = 2, 3, ..., 40.

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

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

Modify the selection sort function developed in this chapter so that it accepts a second optional argument, which may be either ' up ' or "down'. If the argument is 'up', sort the data in ascending order. If the argument is 'dewn', sort the data in descending order. If the argument is missing, the default case is to sort the data in ascending order. (Be sure to handle the case of invalid arguments, and be sure to include the proper help information in your function.)

Dice Simulation. It is often useful to be able to simulate the throw of a fair die. Write a MATLAB function dice that simulates the throw of a fair die by returning some random integer between 1 and 6 every time that it is called. (Hint: Call randomo to generate a randem number. Divide the possible values out of random0 into six equal intervals, and return the number of the interval that a given random value falls into.)

Rayleigh Distribution. The Rayleigh distribution is another random number distribution that appears in many practical problems. A Rayleighdistributed random value can be created by taking the square root of the sum of the squares of two normally distributed random values. In other words, to generate a Rayleigh-distributed random value \(r\) get two normally distributed random values \(\left(n_{1}\right.\) and \(\left.n_{2}\right)\), and perform the following calculation. $$ r=\sqrt{n_{1}^{2}+n_{2}^{2}} $$ a. Create a function rayleigh \((\mathrm{n}, \mathrm{m})\) that returns an \(\mathrm{n} \mathrm{x} \mathrm{m}\) array of Rayleigh-distributed random numbers. If only one argument is supplied [rayleigh \((\mathrm{n})\) ], the function should return an \(\mathrm{n} \times \mathrm{n}\) array of Rayleigh- distributed random numbers. Be sure to design your function with input argument checking and with proper documentation for the MATLAB help system. b. Test your function by creating an array of 20,000 Rayleigh-distributed random values and plotting a histogram of the distribution. What does the distribution look like? c. Determine the mean and standard deviation of the Rayleigh distribution.

Derivative in the Presence of Noise. We will now explore the effects of input noise on the quality of a numerical derivative. First, generate an input vector containing 100 values of the function \(\sin x\) starting at \(x=0\), using a step size \(\Delta x\) of \(0.05\), just as you did in the previous problem. Next, use function randomo to generate a small amount of random noise with a maximum amplitude of \(\pm 0.02\), and add that random noise to the samples in your input vector (see Figure 5.8). Note that the peak amplitude of the noise is only \(2 \%\) of the peak amplitude of your signal, since the maximum value of \(\sin x\) is 1. Now take the derivative of the function using the derivative function that you developed in the last problem. How close to the theoretical value of the derivative did you come?

Linear Least-Squares Fit. Develop a function that will calculate slope \(m\) and intercept \(b\) of the least-squares line that best fits an input data set. The input data points \((x, y)\) will be passed to the function in two input arrays, \(x\) and \(y\). (The equations describing the slope and intercept of the least- squares line are given in Example \(4.7\) in the previous chapter.) Test your function using a test program and the 20-point input data set given in Table 5.2.
See all solutions

Recommended explanations on Psychology Textbooks

Cognition

Read Explanation

Social Context of Behaviour

Read Explanation

Psychological Treatment

Read Explanation

Relationships

Read Explanation

Data Handling and Analysis

Read Explanation

Schizophrenia

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.

Sign-up for free