A carton contains 20 bulbs ,5 of which are defective. The probability that,if a sample of 3 bulbs in chosen at random from the carton, 2 will be defective, is

Options

A. `1//16`
B. `3//64`
C. `9//64`
D. `2//3`

Answer

To solve the problem of finding the probability that, if a sample of 3 bulbs is chosen at random from a carton containing 20 bulbs (5 of which are defective), 2 will be defective, we can follow these steps:

### Step 1: Determine the total number of bulbs and the number of defective bulbs.
- Total bulbs = 20
- Defective bulbs = 5
- Non-defective bulbs = Total bulbs - Defective bulbs = 20 - 5 = 15

**Hint:** Identify the total number of items and categorize them into defective and non-defective.

### Step 2: Calculate the probabilities of selecting a defective and a non-defective bulb.
- Probability of selecting a defective bulb (P(D)) = Number of defective bulbs / Total bulbs = 5/20 = 1/4
- Probability of selecting a non-defective bulb (P(N)) = Number of non-defective bulbs / Total bulbs = 15/20 = 3/4

**Hint:** Use the formula for probability, which is the number of favorable outcomes divided by the total number of outcomes.

### Step 3: Determine the different ways to choose 2 defective bulbs and 1 non-defective bulb.
The possible arrangements for selecting 2 defective bulbs (D) and 1 non-defective bulb (N) in a sample of 3 can be represented as:
1. D, D, N
2. D, N, D
3. N, D, D

This gives us a total of 3 arrangements.

**Hint:** Consider the permutations of the selected items to account for different sequences.

### Step 4: Calculate the probability for one arrangement (D, D, N).
Using the probabilities calculated in Step 2:
- Probability of D, D, N = P(D) * P(D) * P(N) = (1/4) * (1/4) * (3/4) = 3/64

**Hint:** Multiply the probabilities of each event occurring in the arrangement.

### Step 5: Multiply the probability of one arrangement by the number of arrangements.
Since there are 3 arrangements, the total probability of getting 2 defective bulbs and 1 non-defective bulb is:
- Total Probability = Number of arrangements * Probability of one arrangement = 3 * (3/64) = 9/64

**Hint:** Use the total number of arrangements to find the overall probability.

### Final Answer:
The probability that, if a sample of 3 bulbs is chosen at random from the carton, 2 will be defective is **9/64**.

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