Suppose three random grade 11 students are inside a classroom. A random variable X represents the number of students who share the same birthday among these students. Construct a probability distribution of X.

To construct a probability distribution of the random variable \( X \), where \( X \) represents the number of students who share the same birthday among three random grade 11 students, we need to consider the possible values of \( X \) and calculate their corresponding probabilities.

### Step 1: Identify Possible Values of \( X \)
The random variable \( X \) can take the following values:
- \( X = 1 \): No two students share the same birthday. (All three students have different birthdays.)
- \( X = 2 \): Exactly two students share the same birthday, while the third student has a different birthday.
- \( X = 3 \): All three students share the same birthday.

### Step 2: Calculate the Probability for Each Case

#### **Case 1: \( X = 1 \)**
No two students share the same birthday. The first student can have any birthday (365 choices). The second student must have a different birthday (364 choices), and the third student must have a different birthday from the first two (363 choices).

The probability is:
\[
P(X = 1) = \frac{365 \times 364 \times 363}{365^3} = \frac{364 \times 363}{365^2} \approx 0.9918
\]

#### **Case 2: \( X = 2 \)**
Exactly two students share the same birthday, and the third student has a different birthday. There are 3 ways to choose which two students share the same birthday. The shared birthday can be any of the 365 days, and the third student must have a different birthday (364 choices).

The probability is:
\[
P(X = 2) = \frac{3 \times 365 \times 364}{365^3} = \frac{3 \times 364}{365^2} \approx 0.0082
\]

#### **Case 3: \( X = 3 \)**
All three students share the same birthday. There is only 1 way this can happen: they all have the same birthday.

The probability is:
\[
P(X = 3) = \frac{365}{365^3} = \frac{1}{365^2} \approx 0.000008
\]

### Step 3: Construct the Probability Distribution
Now that we have the probabilities, the probability distribution of \( X \) is:

| \( X \) | Probability \( P(X) \) |
|:------:|:----------------------:|
|   1    |       \( 0.9918 \)      |
|   2    |       \( 0.0082 \)      |
|   3    |     \( 0.000008 \)     |

### Final Answer:
The probability distribution of \( X \), where \( X \) is the number of students who share the same birthday among three students, is as shown in the table above.

Would you like any further details or have any questions?

### Related Questions:
1. What is the expected value of \( X \)?
2. How would the probability distribution change if there were 4 students?
3. What is the variance of \( X \)?
4. How does the probability of \( X = 2 \) change as the number of students increases?
5. How would the problem change if there were fewer than 365 days in a year?
6. What is the probability that at least two students share the same birthday?
7. How does this problem relate to the "birthday paradox"?
8. Can this distribution be approximated for a larger number of students?

### Tip:
When dealing with problems involving probabilities and birthdays, it's helpful to remember that the number of possible outcomes increases exponentially as the number of students increases, which can lead to surprising results, such as in the birthday paradox.

Explore the probability distribution of a random variable X representing the number of students who share the same birthday among three grade 11 students. Learn how to calculate probabilities for each case where X = 1, X = 2, and X = 3. Detailed solution and analysis provided.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation