Let X have possible values {1, 2, 3, 4, 5} and probability mass func- tion x 1 2 3 4 5 pX (x) 1 7 1 14 3 14 2 7 2 7 (a) Calculate P (X = 3). (b) Calculate P (X < 3). (c) Calculate P( X < 4 | X > 16 /38)

Let's break down the problem step by step:

### Given:
- \( X \) has possible values \( \{1, 2, 3, 4, 5\} \)
- The probability mass function \( p_X(x) \) is as follows:

| \( x \)     | 1   | 2   | 3   | 4   | 5   |
|-------------|-----|-----|-----|-----|-----|
| \( p_X(x) \)| \( \frac{1}{7} \) | \( \frac{1}{14} \) | \( \frac{3}{14} \) | \( \frac{2}{7} \) | \( \frac{2}{7} \) |

#### (a) Calculate \( P(X = 3) \):
This is the probability mass function evaluated at \( X = 3 \):
\[
P(X = 3) = p_X(3) = \frac{3}{14}
\]

#### (b) Calculate \( P(X < 3) \):
This is the probability that \( X \) takes on a value less than 3. The possible values less than 3 are 1 and 2, so we sum the probabilities for \( X = 1 \) and \( X = 2 \):
\[
P(X < 3) = P(X = 1) + P(X = 2) = \frac{1}{7} + \frac{1}{14}
\]
\[
P(X < 3) = \frac{2}{14} + \frac{1}{14} = \frac{3}{14}
\]

#### (c) Calculate \( P(X < 4 \mid X > \frac{16}{38}) \):
First, simplify \( \frac{16}{38} \), which is equivalent to \( \frac{8}{19} \). Since \( X \) can only take the values 1, 2, 3, 4, or 5, and the smallest probability for \( X \) is \( \frac{1}{14} \approx 0.071 \), all values of \( X \) exceed \( \frac{8}{19} \approx 0.421 \).

Therefore, this condition \( X > \frac{16}{38} \) includes all possible values for \( X \). So this conditional probability becomes:
\[
P(X < 4 \mid X > \frac{16}{38}) = P(X < 4)
\]
Now, calculate \( P(X < 4) \), which is the sum of the probabilities for \( X = 1, 2, 3 \):
\[
P(X < 4) = P(X = 1) + P(X = 2) + P(X = 3) = \frac{1}{7} + \frac{1}{14} + \frac{3}{14}
\]
\[
P(X < 4) = \frac{2}{14} + \frac{1}{14} + \frac{3}{14} = \frac{6}{14} = \frac{3}{7}
\]

Thus, the conditional probability is:
\[
P(X < 4 \mid X > \frac{16}{38}) = \frac{3}{7}
\]

### Final Answers:
- (a) \( P(X = 3) = \frac{3}{14} \)
- (b) \( P(X < 3) = \frac{3}{14} \)
- (c) \( P(X < 4 \mid X > \frac{16}{38}) = \frac{3}{7} \)

Would you like more details or have any questions? Here are 5 related questions to extend the discussion:

1. How do we compute conditional probabilities in general?
2. What is the total probability for all possible values of \( X \)?
3. Can you calculate the expectation \( E(X) \) using this probability mass function?
4. How would the calculations change if one of the probabilities were adjusted?
5. What is the significance of \( X > \frac{16}{38} \) in this case?

**Tip:** Always verify that the sum of probabilities from a probability mass function (PMF) equals 1 to ensure it's valid.

Let X have possible values {1, 2, 3, 4, 5} and probability mass function x 1 2 3 4 5 pX(x) 1/7 1/14 3/14 2/7 2/7 (a) Calculate P(X = 3). (b) Calculate P(X < 3). (c) Calculate P(X < 4 | X > 16/38).

Solve a probability problem involving the discrete random variable X with values {1, 2, 3, 4, 5}. Learn how to calculate P(X = 3), P(X < 3), and the conditional probability P(X < 4 | X > 16/38) using probability mass functions. Suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation