A box contains 100 bolts and 50 nuts, it is given that 50% bolts and 50% nuts are rusted. Two objects are selected from the box at random. Find the probability that both are bolts or both are rusted.

Answer

To solve the problem, we need to find the probability that both selected objects are either bolts or both are rusted. We can break this down into two events:

- Event A: Selecting two rusted items.
- Event B: Selecting two bolts.

We will also need to calculate the intersection of these two events, which is the probability of selecting two rusted bolts (Event A ∩ B).

### Step 1: Determine the total number of items and rusted items

1. **Total bolts** = 100
2. **Total nuts** = 50
3. **Total items** = 100 bolts + 50 nuts = 150 items

4. **Rusty bolts** = 50% of 100 = 50
5. **Rusty nuts** = 50% of 50 = 25
6. **Total rusty items** = 50 rusty bolts + 25 rusty nuts = 75 rusty items

### Step 2: Calculate the probabilities

#### Probability of Event A (Selecting two rusted items)

The probability of selecting two rusted items can be calculated using combinations:

\[ P(A) = \frac{\text{Number of ways to choose 2 rusted items}}{\text{Total ways to choose 2 items}} \]

- Number of ways to choose 2 rusted items from 75:
\[ \binom{75}{2} = \frac{75 \times 74}{2} = 2775 \]

- Total ways to choose 2 items from 150:
\[ \binom{150}{2} = \frac{150 \times 149}{2} = 11175 \]

Thus, the probability of Event A is:
\[ P(A) = \frac{2775}{11175} \]

#### Probability of Event B (Selecting two bolts)

The probability of selecting two bolts can also be calculated using combinations:

\[ P(B) = \frac{\text{Number of ways to choose 2 bolts}}{\text{Total ways to choose 2 items}} \]

- Number of ways to choose 2 bolts from 100:
\[ \binom{100}{2} = \frac{100 \times 99}{2} = 4950 \]

Thus, the probability of Event B is:
\[ P(B) = \frac{4950}{11175} \]

#### Probability of Event A ∩ B (Selecting two rusted bolts)

Now we need to calculate the probability of selecting two rusted bolts:

- Number of rusty bolts = 50
- Number of ways to choose 2 rusty bolts from 50:
\[ \binom{50}{2} = \frac{50 \times 49}{2} = 1225 \]

Thus, the probability of Event A ∩ B is:
\[ P(A \cap B) = \frac{1225}{11175} \]

### Step 3: Use the formula for the probability of A or B

Using the formula for the probability of the union of two events:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

Substituting the values we calculated:
\[ P(A \cup B) = \frac{2775}{11175} + \frac{4950}{11175} - \frac{1225}{11175} \]

Calculating this gives:
\[ P(A \cup B) = \frac{2775 + 4950 - 1225}{11175} = \frac{6500}{11175} \]

### Step 4: Simplify the probability

Now we can simplify \(\frac{6500}{11175}\):
- Dividing both numerator and denominator by 25 gives:
\[ P(A \cup B) = \frac{260}{447} \]

### Final Answer

The probability that both selected objects are either bolts or both are rusted is:
\[ P(A \cup B) = \frac{260}{447} \approx 0.58 \]