https://file.aiquickdraw.com/mathbot/file/17290347064146xjcsebh.jpg

Let's solve the given problem based on the probability distribution table you provided:

| **x** | **P(x)**  |
|-------|-----------|
| 0     | 0         |
| 1     | 0.12      |
| 2     | 0.20      |
| 3     | 0.34      |
| 4     | 0.17      |
| 5     | 0.13      |
| 6     | 0.04      |

We need to compute the following probabilities:

---

### a. \( P(4 \leq X \leq 5) \)
This asks for the probability that \( X \) is between 4 and 5, inclusive.  
\[
P(4 \leq X \leq 5) = P(X = 4) + P(X = 5)
\]
\[
P(4 \leq X \leq 5) = 0.17 + 0.13 = 0.30
\]

Answer: **0.30**

---

### b. \( P(2 \leq X < 4) \)
This asks for the probability that \( X \) is between 2 and 4, where 2 is included, but 4 is not.  
\[
P(2 \leq X < 4) = P(X = 2) + P(X = 3)
\]
\[
P(2 \leq X < 4) = 0.20 + 0.34 = 0.54
\]

Answer: **0.54**

---

### c. \( P(X \leq 5) \)
This asks for the cumulative probability that \( X \) takes any value from 0 to 5, inclusive.  
\[
P(X \leq 5) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5)
\]
\[
P(X \leq 5) = 0 + 0.12 + 0.20 + 0.34 + 0.17 + 0.13 = 0.96
\]

Answer: **0.96**

---

If anything is unclear or you'd like more detailed steps, feel free to ask!

---

### 5 Related Questions:
1. What is the total probability sum for any valid probability distribution?
2. How would the probabilities change if the die was fair (uniform distribution)?
3. How do you calculate \( P(3 < X \leq 6) \) using this table?
4. Can we treat this die as biased, and how does that affect expected value calculations?
5. What is the expected value \( E(X) \) for this distribution?

---

**Tip:** Always make sure the sum of all probabilities in a distribution equals 1 to confirm it’s valid.

An unfair die looks like an ordinary six-sided die, but the outcomes are not equally likely. Given a probability distribution, calculate P(4 ≤ X ≤ 5), P(2 ≤ X < 4), and P(X ≤ 5).

Learn how to calculate specific probabilities using an unfair die's probability distribution. This example includes finding P(4 ≤ X ≤ 5), P(2 ≤ X < 4), and cumulative probabilities P(X ≤ 5). Suitable for Grades 9-12 students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation