What is the probability that at least two persons out of a group of three persons were born in the same month (disregard year)?

Options

A. `(33)/(144)`
B. `(17)/(72)`
C. `(1)/(144)`
D. `(2)/(9)`

Explanation

To solve the problem of finding the probability that at least two persons out of a group of three were born in the same month, we can use the complementary probability approach. Here’s the step-by-step solution:

### Step 1: Understand the Problem
We need to find the probability that at least two out of three people share the same birth month. Instead of calculating this directly, we will first calculate the probability that all three people are born in different months and then subtract this from 1.

### Step 2: Total Number of Months
There are 12 months in a year.

### Step 3: Calculate the Total Cases
The total number of ways to assign birth months to three people, where each can be born in any of the 12 months, is given by:
\[ 12^3 \]
This is because each of the three people can independently be born in any of the 12 months.

### Step 4: Calculate the Favorable Cases (All Different Months)
To find the number of ways in which all three people are born in different months, we can select 3 months from 12 and then assign these months to the 3 people. The number of ways to choose 3 different months from 12 is given by:
\[ 12 \times 11 \times 10 \]
This is because:
- The first person can be born in any of the 12 months.
- The second person can be born in any of the remaining 11 months.
- The third person can be born in any of the remaining 10 months.

Thus, the total number of favorable cases (where all three have different birth months) is:
\[ 12 \times 11 \times 10 = 1320 \]

### Step 5: Calculate the Probability of All Different Months
Now, we can calculate the probability that all three people are born in different months:
\[ P(\text{all different}) = \frac{\text{Number of favorable cases}}{\text{Total cases}} = \frac{1320}{12^3} = \frac{1320}{1728} \]

### Step 6: Simplify the Probability
To simplify \(\frac{1320}{1728}\), we can divide both the numerator and the denominator by their greatest common divisor (GCD):
\[ \frac{1320 \div 12}{1728 \div 12} = \frac{110}{144} = \frac{55}{72} \]

### Step 7: Calculate the Required Probability
Now, we find the probability that at least two people share the same birth month:
\[ P(\text{at least two same}) = 1 - P(\text{all different}) = 1 - \frac{55}{72} = \frac{72 - 55}{72} = \frac{17}{72} \]

### Final Answer
Thus, the probability that at least two persons out of a group of three were born in the same month is:
\[ \frac{17}{72} \]