Question

The probability that the birthdays of 4 different persons will fall in exactly two calendar months is : (a) \(\frac{77}{1728}\) (b) \(\frac{17}{87}\) (c) \(\frac{11}{144}\) (d) none of these

Short answer

Answer: (a) $\frac{77}{1728}$.

Step-by-step solution

Determine the total number of combinations

There are 12 months in a year. The birthdays of 4 different persons can fall in any of these 12 months. So, there are a total of \(12^4\) possible combinations of months for the 4 people's birthdays.

Count the favorable outcomes

We want the birthdays of the 4 persons to fall in exactly two months. We can first choose those two months in \(\binom{12}{2}\) ways. Now, we can distribute the 4 birthdays among the two chosen months in the following ways: - 1 birthday in Month A and 3 birthdays in Month B - 2 birthdays in Month A and 2 birthdays in Month B - 3 birthdays in Month A and 1 birthday in Month B There are a total of 3 ways to distribute the 4 birthdays among the two chosen months. Hence, the total favorable outcomes are \(\binom{12}{2} \times 3\).

Calculate the probability

Now, we can calculate the probability by dividing the favorable outcomes by the total possible combinations: Probability \(= \frac{\binom{12}{2} \times 3}{12^4}\) Using the binomial coefficient formula and simplifying, we get: Probability \(= \frac{(12 \times 11) \times \frac{1}{2} \times 3}{12^4} = \frac{198}{12^3} = \frac{77}{1728}\) Therefore, the probability that the birthdays of 4 different persons will fall in exactly two calendar months is \(\frac{77}{1728}\). So, the correct answer is (a).

Key concepts

Combinatorics

Combinatorics is a branch of mathematics focused on counting, arranging, and finding ways to combine different elements of a set. It is all about figuring out how many different ways things can happen. In this problem, we use combinatorics to count the different ways that the birthdays of four people can fall across two different months.

• When we talk about combinations, we mean ways of selecting items from a group, where the order does not matter.
• In the exercise, every choice of two months from the 12 available months is considered a combination.

To represent this concept numerically, we use something called the binomial coefficient. The combination formula is often represented with a notation like \( \binom{n}{k} \), where \( n \) is the total number of items that you can choose from and \( k \) is the number of items that you choose. Simply put, the goal is to calculate the number of ways to choose 2 months out of 12 available months, without caring about the order of selection.

Birthday Problem

The Birthday Problem is a classic example in probability theory, which demonstrates some counter-intuitive results. It explores the likelihood that, in a group of people, some will have the same birthday.

• This is not exactly about finding matching birthdays but about seeing how birthdays can be distributed among months.
• In our specific scenario, we are examining how the birthdays of 4 individuals can fall into only two different months.

Though at first glance it may seem improbable, the mathematical approach used demonstrates how considering different distribution scenarios helps determine the probability of such events occurring. It's these distribution patterns—such as having all birthdays condensed into certain months—that the exercise highlights.

Binomial Coefficient

At the core of solving this problem is the binomial coefficient, represented by the expression \( \binom{n}{k} \). It is a way to determine how many combinations of \( k \) items can be selected from a larger collection of \( n \) items. In the context of the given exercise, the formula \( \binom{12}{2} \) is crucial. Here's why:

• This expression calculates the number of ways to select 2 months from 12 months.
• The simplification of \( \binom{12}{2} = \frac{12 \times 11}{2} = 66 \) reveals that there are 66 possible pairs of months that could be selected.

the binomial coefficient gives us the foundational count needed for further calculations. Without understanding this, figuring out the overall probability of having the birthdays split between these months would be very hard.

Mathematical Problem Solving

Mathematical problem-solving is about creating a structured approach to tackle complex questions, like the one in this exercise. By using a step-by-step method, we can easily break down and analyze the problem. Here's a general outline:* Every solution starts by defining the total number of possible outcomes. Here, it is calculated as \( 12^4 \), representing all possible ways 4 birthdays can fall into 12 months.* Next, identify favorable outcomes. In this problem, it's about finding how the selected months can accommodate the birthdays.This kind of structure is widely used:

• Step 1: Understand the event's total possible outcomes.
• Step 2: Identify desirable outcomes matching the problem statement.
• Step 3: Complete the calculation of probability using basic fraction operations.

By organizing the problem into these stages, we simplify the task, making it approachable and solvable step-by-step.