Three archers aim at the same target, each shooting once and independently. Their chances of hitting are 1/2, 1/3 and 1/4. What is the probability that the target is hit at least once?

Question 47

HardOriginal question

Three archers aim at the same target, each shooting once and independently. Their chances of hitting are $\frac{1}{2}$ , $\frac{1}{3}$ and $\frac{1}{4}$ . What is the probability that the target is hit at least once?

\frac{1}{24}

Correct answerYour answer

\frac{3}{4}

Correct answerYour answer

\frac{11}{24}

Correct answerYour answer

\frac{1}{4}

Correct answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer B

Correct answer: B

The target escapes only if all three miss: $\frac{1}{2} \times \frac{2}{3} \times \frac{3}{4} = \frac{6}{24} = \frac{1}{4}$ .
P(hit at least once) $= 1 - \frac{1}{4} = \frac{3}{4}$ .

Worth noting

Adding the three chances gives $\frac{13}{12}$ , more than 1, which is impossible. $\frac{1}{24}$ is P(all hit), $\frac{1}{4}$ is P(all miss) and $\frac{11}{24}$ is P(exactly one hits).

Concept: 'At least one' for three independent events
Typical timeTypical: 120 s
Your timeYours: —

The rule this testsBasic probability

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Ten questions from Counting and Basic Probability, at your own pace.

Three archers aim at the same target, each shooting once and independently. Their chances of hitting are 12\frac{1}{2}21​, 13\frac{1}{3}31​ and 14\frac{1}{4}41​. What is the probability that the target is hit at least once?

Explanation

Three archers aim at the same target, each shooting once and independently. Their chances of hitting are $\frac{1}{2}$ , $\frac{1}{3}$ and $\frac{1}{4}$ . What is the probability that the target is hit at least once?