Question

A hunter is firing at a target. He has only 10% chance of hitting it in one round. The number of rounds he must fire in order to have at least 50% chance of hitting the target at least once, is

• A. 8
• B. 7
• C. 6
• D. 5

Hint:

When dealing with "at least one" probability problems, it is almost always easier to calculate the probability of the complement event (hitting zero times) and subtract it from 1.

Answer 1

The Correct Option is B

Concept: Let $p$ be the probability of hitting the target in one round, so $p = 0.1$. The probability of missing the target in one round is $q = 1 - p = 0.9$. For $n$ rounds, the probability of missing the target every time is $q^n$. The probability of hitting the target at least once is $1 - q^n$.

Step 1: Set up the inequality based on the condition that the probability of hitting the target at least once is at least 50%. \[ 1 - (0.9)^n \geq 0.5 \]

Step 2: Simplify the inequality. \[ 0.5 \geq (0.9)^n \] \[ (0.9)^n \leq 0.5 \]

Step 3: Solve for the smallest integer \(n\) by testing values.
For $n = 5$: $(0.9)^5 = 0.59049$ (which is $> 0.5$)
For $n = 6$: $(0.9)^6 \approx 0.5314$ (which is $> 0.5$)
For $n = 7$: $(0.9)^7 \approx 0.4783$ (which is $< 0.5$)

Step 4: Conclusion based on the inequality.
Since $(0.9)^7 \leq 0.5$, the hunter must fire at least 7 rounds. center minipage0.35

Number of rounds (n) = 7 minipage center