Assignment: 1 whole sheet of paper
Suppose three test kits are tested at random. Let represent the
defective test kit and let represent the non-defective test kit. If we
let be the random variable for the number of defective test kits,
construct the probability distribution of the random variable .

Ask by Wood Washington. in the Philippines

Jan 06,2025

Question

Assignment: 1 whole sheet of paper
Suppose three test kits are tested at random. Let represent the
defective test kit and let represent the non-defective test kit. If we
let be the random variable for the number of defective test kits,
construct the probability distribution of the random variable .

Ask by Wood Washington. in the Philippines

Jan 06,2025

UpStudy AI · Accepted Answer

Certainly! Let's construct the probability distribution for the random variable $$ X $$, which represents the number of defective test kits out of three tested.

### **1. Define the Problem**

- **Total Test Kits Tested:** 3
- **Possible Outcomes for Each Test Kit:**
  - **Defective (D)**
  - **Non-Defective (N)**
- **Random Variable $$ X $$:** Number of defective test kits in the three tested.

### **2. Identify Possible Values of $$ X $$**

Since each test kit can either be defective or not, the possible values for $$ X $$ are:
$$ X = 0, 1, 2, 3 $$

### **3. Determine the Probability for Each Value of $$ X $$**

Assume that each test kit has:
- **Probability of being defective:** $$ p $$
- **Probability of being non-defective:** $$ 1 - p $$

The tests are **independent**, meaning the outcome of one test does not affect the others.

#### **a. $$ X = 0 $$ (No Defective Test Kits)**

All three test kits are non-defective.
$$
P(X = 0) = (1 - p)^3
$$

#### **b. $$ X = 1 $$ (Exactly One Defective Test Kit)**

One test kit is defective, and the other two are non-defective. There are $$ \binom{3}{1} = 3 $$ ways this can happen.
$$
P(X = 1) = 3 \times p \times (1 - p)^2
$$

#### **c. $$ X = 2 $$ (Exactly Two Defective Test Kits)**

Two test kits are defective, and one is non-defective. There are $$ \binom{3}{2} = 3 $$ ways this can happen.
$$
P(X = 2) = 3 \times p^2 \times (1 - p)
$$

#### **d. $$ X = 3 $$ (All Three Test Kits Are Defective)**

All three test kits are defective.
$$
P(X = 3) = p^3
$$

### **4. Summary of the Probability Distribution**

| $$ X $$ | Number of Defective Test Kits | Probability $$ P(X) $$                    |
|---------|-------------------------------|-------------------------------------------|
| 0       | 0                             | $$ (1 - p)^3 $$                           |
| 1       | 1                             | $$ 3p(1 - p)^2 $$                         |
| 2       | 2                             | $$ 3p^2(1 - p) $$                         |
| 3       | 3                             | $$ p^3 $$                                 |

### **5. Example Calculation**

If, for instance, the probability that a single test kit is defective is $$ p = 0.2 $$, then:

- $$ P(X = 0) = (1 - 0.2)^3 = 0.512 $$
- $$ P(X = 1) = 3 \times 0.2 \times (1 - 0.2)^2 = 0.384 $$
- $$ P(X = 2) = 3 \times (0.2)^2 \times (1 - 0.2) = 0.096 $$
- $$ P(X = 3) = (0.2)^3 = 0.008 $$

### **6. Verification**

To ensure the probability distribution is valid, the sum of all probabilities should equal 1:
$$
P(X=0) + P(X=1) + P(X=2) + P(X=3) = (1 - p)^3 + 3p(1 - p)^2 + 3p^2(1 - p) + p^3 = 1
$$
Which holds true for any valid probability $$ p $$ between 0 and 1.

---

**This completes the construction of the probability distribution for the random variable $$ X $$.**
The probability distribution for the number of defective test kits $$ X $$ is as follows: - $$ P(X = 0) = (1 - p)^3 $$ - $$ P(X = 1) = 3p(1 - p)^2 $$ - $$ P(X = 2) = 3p^2(1 - p) $$ - $$ P(X = 3) = p^3 $$ Where $$ p $$ is the probability that a single test kit is defective.

Assignment: 1 whole sheet of paper
Suppose three test kits are tested at random. Let represent the
defective test kit and let represent the non-defective test kit. If we
let be the random variable for the number of defective test kits,
construct the probability distribution of the random variable .

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Assignment: 1 whole sheet of paper Suppose three test kits are tested at random. Let represent the defective test kit and let represent the non-defective test kit. If we let be the random variable for the number of defective test kits, construct the probability distribution of the random variable .

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Assignment: 1 whole sheet of paper
Suppose three test kits are tested at random. Let represent the
defective test kit and let represent the non-defective test kit. If we
let be the random variable for the number of defective test kits,
construct the probability distribution of the random variable .