Ared box contains eight itmes, of which three are defective, and a blue box contains five
itmes, of which two are defective. An item is drawn at random from each box. What is the
probability that both items will be nondefective?

Options

A. `3/20`
B. `3/8`
C. `5/13`
D. `8/13`

Explanation

To solve the problem, we need to find the probability that both items drawn from the red and blue boxes are non-defective. Here’s a step-by-step solution:

### Step 1: Identify the total items and defective items in each box
- **Red Box**: Total items = 8, Defective items = 3
- **Blue Box**: Total items = 5, Defective items = 2

### Step 2: Calculate the number of non-defective items in each box
- Non-defective items in the Red Box = Total items - Defective items = 8 - 3 = 5
- Non-defective items in the Blue Box = Total items - Defective items = 5 - 2 = 3

### Step 3: Calculate the probability of drawing a non-defective item from each box
- Probability of drawing a non-defective item from the Red Box (P(A')) = Number of non-defective items / Total items = 5 / 8
- Probability of drawing a non-defective item from the Blue Box (P(B')) = Number of non-defective items / Total items = 3 / 5

### Step 4: Since the events are independent, calculate the combined probability
- The probability that both items drawn are non-defective is given by the product of the individual probabilities:

\[ P(A' \cap B') = P(A') \times P(B') = \left(\frac{5}{8}\right) \times \left(\frac{3}{5}\right) \]

### Step 5: Perform the multiplication
- Calculate:

\[ P(A' \cap B') = \frac{5 \times 3}{8 \times 5} = \frac{15}{40} \]

### Step 6: Simplify the fraction
- Simplifying \(\frac{15}{40}\):

\[ \frac{15}{40} = \frac{3}{8} \]

### Final Answer
The probability that both items drawn will be non-defective is \(\frac{3}{8}\).

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