In a bombing attack , there is 50% chance that a bomb will hit target . At least two independent hits are required to destroy the target completely. Then the minimum number of bombs, that must be dropped to ensure that there is at least 90% chance of completely destroying the target , is ________

Answer

To solve the problem, we need to determine the minimum number of bombs \( n \) that must be dropped to ensure at least a 90% chance of completely destroying the target, given that each bomb has a 50% chance of hitting the target.

### Step-by-step Solution:

1. **Define the probabilities**:
- Let \( P \) be the probability of a bomb hitting the target, which is \( P = 0.5 \).
- Let \( Q \) be the probability of a bomb not hitting the target, which is \( Q = 1 - P = 0.5 \).

2. **Determine the required condition**:
- We need at least 2 hits to destroy the target. Therefore, we want the probability of getting at least 2 hits to be at least 90%.
- Mathematically, this can be expressed as:
\[ P(X \geq 2) \geq 0.9 \]
- This can be rewritten using the complement:
\[ 1 - P(X < 2) \geq 0.9 \]
- Which simplifies to:
\[ P(X < 2) \leq 0.1 \]

3. **Calculate \( P(X < 2) \)**:
- The event \( X < 2 \) includes the cases where \( X = 0 \) and \( X = 1 \):
\[ P(X < 2) = P(X = 0) + P(X = 1) \]
- Using the binomial distribution:
\[ P(X = k) = \binom{n}{k} P^k Q^{n-k} \]
- Therefore:
\[ P(X = 0) = \binom{n}{0} (0.5)^0 (0.5)^n = (0.5)^n \]
\[ P(X = 1) = \binom{n}{1} (0.5)^1 (0.5)^{n-1} = n \cdot (0.5)^n \]
- Thus:
\[ P(X < 2) = (0.5)^n + n \cdot (0.5)^n = (0.5)^n (1 + n) \]

4. **Set up the inequality**:
- We need:
\[ (0.5)^n (1 + n) \leq 0.1 \]
- Rearranging gives:
\[ 2^n (1 + n) \geq 10 \]

5. **Trial and error to find the minimum \( n \)**:
- Let's test values for \( n \):
- For \( n = 6 \):
\[ 2^6 (1 + 6) = 64 \cdot 7 = 448 \quad (\text{which is } > 10) \]
- For \( n = 7 \):
\[ 2^7 (1 + 7) = 128 \cdot 8 = 1024 \quad (\text{which is } > 10) \]
- For \( n = 5 \):
\[ 2^5 (1 + 5) = 32 \cdot 6 = 192 \quad (\text{which is } > 10) \]
- For \( n = 4 \):
\[ 2^4 (1 + 4) = 16 \cdot 5 = 80 \quad (\text{which is } > 10) \]
- For \( n = 3 \):
\[ 2^3 (1 + 3) = 8 \cdot 4 = 32 \quad (\text{which is } > 10) \]
- For \( n = 2 \):
\[ 2^2 (1 + 2) = 4 \cdot 3 = 12 \quad (\text{which is } > 10) \]
- For \( n = 1 \):
\[ 2^1 (1 + 1) = 2 \cdot 2 = 4 \quad (\text{which is } < 10) \]

6. **Conclusion**:
- The minimum number of bombs that must be dropped to ensure at least a 90% chance of destroying the target is \( n = 7 \).

### Final Answer:
The minimum number of bombs that must be dropped is **7**.