The probability of a shooter hitting a target is $\frac{3}{4}$. How many minimum number of times must he/she fire so that the probability of hitting the target at least once is more than 0.99?

Question

Accepted Answer

To determine the minimum number of times the shooter must fire to ensure the probability of hitting the target at least once is more than 0.99, we need to consider the complementary probability of not hitting the target at all.

The probability of the shooter hitting the target in a single shot is $\frac{3}{4}$. Therefore, the probability of missing the target in a single shot is:

$$ 1 - \frac{3}{4} = \frac{1}{4} $$

If the shooter fires $ n $ times, the probability of missing the target every time is:

$$ \left(\frac{1}{4}\right)^n $$

We want the probability of hitting the target at least once to be more than 0.99. Therefore, the probability of missing the target every time should be less than 0.01:

$$ \left(\frac{1}{4}\right)^n < 0.01 $$

To find the smallest $ n $ that satisfies this inequality, we can calculate:

1. For $ n = 1 $:
   $$ \left(\frac{1}{4}\right)^1 = \frac{1}{4} = 0.25 $$
   This is not less than 0.01.

2. For $ n = 2 $:
   $$ \left(\frac{1}{4}\right)^2 = \frac{1}{16} = 0.0625 $$
   This is not less than 0.01.

3. For $ n = 3 $:
   $$ \left(\frac{1}{4}\right)^3 = \frac{1}{64} \approx 0.015625 $$
   This is not less than 0.01.

4. For $ n = 4 $:
   $$ \left(\frac{1}{4}\right)^4 = \frac{1}{256} \approx 0.00390625 $$
   This is less than 0.01.

Thus, the minimum number of times the shooter must fire to ensure the probability of hitting the target at least once is more than 0.99 is 4. Therefore, Option 2 is correct.

**Explanation for Other Options:**

- **Option 1 (3 shots):** As calculated, firing 3 times results in a probability of missing the target every time as approximately 0.015625, which is not less than 0.01. Therefore, this option is incorrect.

- **Option 3 (1 shot):** Firing only once results in a probability of missing the target as 0.25, which is far greater than 0.01. Thus, this option is incorrect.

- **Option 4 (2 shots):** Firing twice results in a probability of missing the target every time as 0.0625, which is not less than 0.01. Hence, this option is incorrect.

In conclusion, the correct answer is Option 2, as firing 4 times ensures the probability of hitting the target at least once is more than 0.99.