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

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 \( X \).**