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Chapter 1: Problem 16

Suppose there are 30 people at a party. Do you think any two share the same birthday? Let's use the random-number table to simulate the birthdays of the 30 people at the party. Ignoring leap year, let's assume that the year has 365 days. Number the days, with 1 representing January 1,2 representing January \(2,\) and so forth, with 365 representing December \(31 .\) Draw a random sample of 30 days (with replacement). These days represent the birthdays of the people at the party. Were any two of the birthdays the same? Compare your results with those obtained by other students in the class. Would you expect the results to be the same or different?

Short Answer

Expert verified
We expect that at least two people will share a birthday due to the birthday paradox, despite random variance.

Step by step solution

01

Understand the Problem

The problem asks us to simulate the birthdays of 30 people at a party and determine if any two share the same birthday. This is a classic example of the "birthday paradox." We will assume 365 days in the year, ignoring leap years.
02

Set Up the Simulation

We will simulate the selection of birthdays by drawing 30 random numbers, each representing a day in the year (between 1 and 365), using a random-number table or generator. This involves drawing with replacement, meaning each day's number can appear more than once.
03

Conduct the Simulation

Using a random-number generator, generate 30 random integers between 1 and 365. These integers represent the days of birth for the 30 people at the party. For example, you might generate numbers like 45, 100, 200, etc.
04

Check for Matching Birthdays

Examine the list of 30 generated numbers to see if any number appears more than once. If a number occurs multiple times, it indicates that more than one person shares that birthday.
05

Analyze the Result and Theoretical Expectation

Calculate the probabilities involved. Statistically, with 30 people, there's a high probability (about 70%) that at least two people share a birthday, known as the birthday paradox. Compare results with other classmates to observe variations based on different simulations.

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Key Concepts

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

Random Sampling
Random sampling refers to choosing items from a larger set, where each item has an equal chance of being selected. In the context of the birthday paradox exercise, we are selecting random birthdays for a group of 30 people using a random sample.

When we say we draw samples with replacement, it means that each of the 365 possible birthday numbers can appear multiple times in our sample. This free repetition mirrors real-life chances of multiple people having the same birthday at a party.

Random sampling is essential to ensure that every number between 1 and 365 has an equal opportunity to be selected each time we make a draw. This unbiased approach aims to replicate natural occurrences and is foundational in simulation experiments to ensure randomness.
Probability
Probability, in this context, is the measure of the likelihood that a particular event will occur. When discussing the birthday paradox, the probability we're interested in is the chance that at least two people out of 30 share the same birthday.

Despite our intuition suggesting otherwise, the probability is surprisingly high. Mathematically, it's about 70% that at least one pair shares a birthday when there are 30 people in the room.

To grasp this better, consider every new person added isn't just comparing dates with one person but with all who's come before. Each time a new person arrives, they bring a new round of comparisons, which boosts the chances of a match even further.
  • With one person, there are no comparisons.
  • Add another, and that's one comparison, with a low probability of a match.
  • Increase to 30 people, and the number of comparisons and hence potential matches rises dramatically.
Simulation
Simulation is a powerful technique that mimics the outcomes of real-world processes by experimenting on a model rather than through direct observation or testing. The birthday paradox problem in question leverages simulation to explore theoretical probabilities through practical trials.

In the exercise, you conduct a simulation by generating random samples of birthdays. Utilize a random-number generator to emulate the random distribution of these birthdays among the 30 party-goers.

By simulating multiple times, you can observe the frequencies with which birthdays are the same. These experiments help solidify understanding, visualizing how probabilities play out in repeated trials. Even though each individual simulation may yield different outcomes (sometimes two shared birthdays, sometimes not), collectively they reinforce the statistical probability expectation.
  • Simulations provide hands-on insight into theoretical concepts.
  • They allow us to compare practical results with calculated probabilities.
  • Many trials give a clearer picture of randomness at work.

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Most popular questions from this chapter

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