Question

A math class has 25 students. Assuming that all of the students were born in the first half of the year-between January 1st and June 30 th \(-\) what is the probability that at least two students have the same birthday? Assume that nobody was born on leap day.

Short answer

The probability is approximately 56.6% that at least two students share a birthday.

Step-by-step solution

Understand the Birthday Paradox

The problem you're dealing with is a variant of the birthday paradox. Instead of the usual 365 days, we have only 181 days to work with due to the restriction to the first half of the year. The goal is to find the probability that at least two students share the same birthday.

Find Total Days

Identify the total number of days in the first half of the year. Since we exclude leap day, there are 31 (January) + 28 (February) + 31 (March) + 30 (April) + 31 (May) + 30 (June) = 181 days.

Calculate Probability of No Shared Birthday

Start by finding the probability that no two students share a birthday. For each student having a distinct birthday, the probability is calculated as: \[ P_{ ext{no shared birthday}} = \frac{181}{181} \times \frac{180}{181} \times \frac{179}{181} \times \ldots \times \frac{157}{181} \]This factors in the number of ways to choose distinct birthdays for each student.

Calculate Probability of At Least One Shared Birthday

The probability that at least two students share a birthday is the complement of the event that no students share a birthday. Thus, \[ P_{ ext{at least one shared birthday}} = 1 - P_{ ext{no shared birthday}} \]Substitute the calculated value from Step 3 into this formula.

Compute the Solution

Calculate the final probability using the formula derived above. Using a calculator or software slip-in tool:\[ P_{ ext{no shared birthday}} \approx 0.434 \]Thus, the probability that at least two students share a birthday is:\[ P_{ ext{at least one shared birthday}} \approx 1 - 0.434 = 0.566 \]

Key concepts

birthday paradox

The birthday paradox is a fascinating concept in probability theory that often leads to surprising results. It poses the question: "In a group of people, what is the likelihood that at least two individuals share the same birthday?" Although it might seem intuitive to think that the probability of shared birthdays is low, the paradox reveals otherwise. It only takes a group of 23 people for there to be more than a 50% chance that two of them share a birthday, given a typical 365-day year.
In the context of the exercise, the birthday paradox is modified to consider only the first six months of the year, totaling 181 days. This alteration changes the dynamics but maintains the core concept. The intrigue lies in how quickly the probability of shared birthdays increases as the group size grows, even with fewer days to consider.
Thus, the birthday paradox, whether in a typical or modified scenario, highlights our common misunderstanding of probability and illustrates the non-intuitive nature of probability theory.

combinatorics

Combinatorics is a branch of mathematics focused on counting, combination, and permutation of sets. It enables the calculation of probabilities by understanding the myriad ways in which events can occur.
In the birthday paradox problem, combinatorics comes into play when calculating the probability of no shared birthdays among students. Imagine choosing birthdays for each student, where the birthday of each subsequent student must differ from those already chosen.
To simplify, consider just the first student and the 181 possible days for their birthday. The second student would then have 180 choices, the third 179, and so on. Therefore, combinatorics helps us compute these probabilities by considering all possible arrangements and restricting those that meet certain conditions, such as no shared birthdays.

• Calculate the number of available choices for each student.
• Determine the conditions for non-shared birthdays.
• Account for overlapping possibilities, if any.

By understanding these principles, we gain better insight into how seemingly simple questions can have complex, yet calculable, solutions.

probability calculation

Probability calculation is a crucial step in tackling the birthday paradox. It involves determining the likelihood of specific outcomes under certain conditions. In this exercise, we want to calculate the probability that at least two students share the same birthday out of 25 students born during the first half of the year.
First, calculate the probability of the more straightforward event: that all students have unique birthdays. This is done by multiplying the probabilities that each subsequent student does not share a birthday with any of the previous ones. For instance, the first student can be born on any of the 181 days, but each subsequent student has one less option to maintain uniqueness.
The formula used is: \[ P_{\text{no shared birthday}} = \frac{181}{181} \times \frac{180}{181} \times \cdots \times \frac{157}{181} \]This gives us the likelihood that no two students share a birthday.
Then, applying the complement rule of probability helps compute the likelihood of at least one shared birthday. This rule states that the probability of "at least one" is 1 minus the probability of "none." Thus: \[ P_{\text{at least one shared birthday}} = 1 - P_{\text{no shared birthday}} \]This simple subtraction gives the desired outcome probability.

• Identify all possible unique outcomes.
• Use successive probability multiplications for distinct events.
• Apply the complement rule for comprehensive results.

Through structured probability calculation, we unravel the surprising results of the birthday paradox.