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Chapter 4: Problem 2

Extend our scaling analysis to the three-birthday problem: How many people must be in a room before the probability of three people sharing a birthday rises above \(0.5\) ? (The results of the exact calculation, along with many approximations, are given by Diaconis and Mosteller [10].)

Short Answer

Expert verified
To find the minimum number of people (n) required in a room for the probability of at least three people sharing a birthday to rise above 0.5, we use the probability formula, iterating until the probability exceeds 0.5. By approximation, this occurs when there are about 88 people in the room. Note that the actual result may vary slightly and require either statistical software or a table of probabilities for a more exact calculation.

Step by step solution

01

Laying out the basic formula

First, calculate the total number of ways three people can be selected from a group of n people. This is given by the combination formula \( C(n, 3) \). Here, \( C(a, b) \) represents the number of combinations of a items taken b at a time and is given by the formula \[ C(a, b) = \frac{{a!}}{{b!(a-b)!}} \] where \( a! \) is the factorial of a, the product of all positive integers up to a. In this context, \( a \) is the total number of people in the room (which we're trying to find) and \( b = 3 \).
02

Calculate the probability

Having defined \( n \) as the number of people in the room, the formula for the probability (P) that at least 3 people share the same birthday is given by the following formula: \[ P = 1 - \frac{{365}}{{365}} \times \frac{{(365 - 1)}}{{365}} \times \frac{{(365 - 2)}}{{365}} \times \ldots \times \frac{{(365 - n + 1)}}{{365}} \] The expression inside the brackets is the probability that n people all have different birthdays.
03

Iterative process

Since there's no direct analytical solution to the above equation when \( P > 0.5 \), use an iterative process. Try to increase the value of \( n \) until \( P > 0.5 \). Start from n=1 and go upwards.
04

Finding the number

By using the process described in Step 3, when using approximation the number could be found to be about 88. That is, if there are 88 people in a room, the probability that at least three of them share a birthday is just over 0.5. Please note: The mathematical calculation involves approximation and the actual result could vary slightly. A more exact calculation would require the use of either statistical software or a table of probabilities for the Birthday problem. This is provided for instructional purposes as a simplified explanation of a complex statistical question.

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Key Concepts

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

Probability Theory
If you've ever planned a surprise party and worried about two guests wearing the same outfit, you've thought about 'probability theory' without even knowing it. In a formal sense, probability theory is a branch of mathematics that deals with calculating the likelihood of various events. It allows us to quantify the unforeseeable, giving us the power to predict events like flipping a coin, rolling dice, or even guessing the number of people needed in a room for a shared birthday.

Considering the birthday problem, probability theory helps us answer the question: 'What are the chances of three people sharing the same birthday in a group?' By analyzing patterns and applying specific formulas related to probability, we can estimate the likelihood of such an event occurring, guiding us to better understand the 'uncertain' and make more informed decisions in our daily lives.
Combinatorics
Combinatorics is an entire playground of mathematics, where the slides and swings are all about counting, arranging, and combining things. For our birthday example, combinatorics is essential in understanding how to count possible combinations of birthdays among a set of people.

Specifically, we use the concept of combinations, which are different groupings of a fixed number of elements (in this case, people) that we can form without regard to the order in which they're arranged. We might think, 'Why does order not matter?' Imagine arranging three people whose birthdays you're trying to match鈥攏o matter how you change their order, the group is still the same. Combinatorics shares its toys well with probability theory to help us calculate just how many different groups of three people we can make from any given room.
Iterative Process
Sometimes, finding an answer isn't as simple as plugging numbers into a formula. This is where an 'iterative process' comes into play, which is a bit like trying on different shoes until you find the one that fits. In mathematics, an iterative process means repeating a series of steps over and over, each time using the result from the last step as the starting point for the next, until we arrive at a satisfactory answer.

For the three-person birthday problem, we incrementally increase the number of people in the room, recalculating the probability each time until we reach our desired probability of over 50%. This approach is a systematic trial-and-error, giving us a practical method to solve problems that might be too complex for a direct solution.
Factorial

The Multiplying March

If you've ever added up how many different handshakes could happen in a room full of people, you've danced with a 'factorial.' The factorial of a number (expressed as '!') is the product of all positive integers leading up to that number. It's like a countdown 鈥� for instance, the factorial of 5 (denoted as 5!) is 5 x 4 x 3 x 2 x 1.

Why is this important for the birthday problem? It allows us to calculate combinations efficiently. Factorials help define the total possible ways birthdays can be distributed among people in the room, a critical part of solving our problem. Grasping this concept not only helps us with birthday problems but also is a key player in various fields of mathematics and science, such as arranging items or understanding permutations.

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Most popular questions from this chapter

In Problem 3.23, about the shortest path that bisects an equilateral triangle, one candidate path is a horizontal line. How long is that line relative to a side of the triangle?

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