A certain type of missile hits the target with probability p= 0.3. What is the least number of missiles should be fired so that there is at least an 80% probability that the target is hit?

Options

A. 5
B. 6
C. 7
D. None of the above

Answer

To solve the problem of determining the least number of missiles that should be fired so that there is at least an 80% probability that the target is hit, we can follow these steps:

### Step 1: Define the probabilities
Let \( p = 0.3 \) be the probability that a missile hits the target. Therefore, the probability that a missile misses the target is \( q = 1 - p = 1 - 0.3 = 0.7 \).

### Step 2: Set up the probability condition
We want the probability of hitting the target at least once when \( n \) missiles are fired to be at least 80%. The probability of hitting the target at least once can be expressed as:
\[ P(\text{at least 1 hit}) = 1 - P(\text{no hits}) \]
The probability of no hits when \( n \) missiles are fired is given by:
\[ P(\text{no hits}) = q^n = (0.7)^n \]
Thus, we can write:
\[ P(\text{at least 1 hit}) = 1 - (0.7)^n \]

### Step 3: Set up the inequality
We want:
\[ 1 - (0.7)^n \geq 0.8 \]
This simplifies to:
\[ (0.7)^n \leq 0.2 \]

### Step 4: Solve the inequality
To find the smallest integer \( n \) that satisfies this inequality, we can take the logarithm of both sides:
\[ \log((0.7)^n) \leq \log(0.2) \]
This simplifies to:
\[ n \cdot \log(0.7) \leq \log(0.2) \]
Since \( \log(0.7) \) is negative, we can divide by it, which reverses the inequality:
\[ n \geq \frac{\log(0.2)}{\log(0.7)} \]

### Step 5: Calculate the values
Now we can calculate the values using a calculator:
- \( \log(0.2) \approx -0.6990 \)
- \( \log(0.7) \approx -0.1553 \)

Thus:
\[ n \geq \frac{-0.6990}{-0.1553} \approx 4.5 \]

### Step 6: Determine the least integer value of \( n \)
Since \( n \) must be an integer, we round up to the next whole number:
\[ n = 5 \]

### Conclusion
The least number of missiles that should be fired so that there is at least an 80% probability that the target is hit is **5**.

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