Question

The probability of a girl hitting a target is \( 1/2 \). How many times must she fire so that the probability of hitting the target at least once is more than \( 90% \)?

• A. 2
• B. 4
• C. 5
• D. 3

Hint:

In 'at least once' probability questions, always solve using the complement event (none of the events occur).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
To find the probability of "at least one success", it is easier to calculate \( 1 - P(\text{no successes}) \).

Step 2: Key Formula or Approach:
Let \( n \) be the number of fires. The probability of missing is \( 1/2 \).
The probability of missing all \( n \) times is \( (1/2)^n \).
We require \( 1 - (1/2)^n > 0.90 \).

Step 3: Detailed Explanation:
\[ 1 - (1/2)^n > 0.90 \]
\[ 0.10 > (1/2)^n \]
\[ (1/2)^n 10 \]
Testing integer values for \( n \):
For \( n = 3 \), \( 2^3 = 8 \) (\( < 10 \)).
For \( n = 4 \), \( 2^4 = 16 \) (\( > 10 \)).
Since \( n=4 \) is the smallest integer where the probability \( 1 - 1/16 = 0.9375 \) exceeds \( 0.90 \), the answer is 4.

Step 4: Final Answer:
She must fire 4 times.