Shoot The Target

Probability Level 2

If I shoot 20 arrows at a target the probability of hitting it at least once is $\dfrac89$ . What is the probability of hitting the target at least once if I shoot 10 arrows?

Note: Assume each shot is independent.

\frac49

\frac23

\frac13

\frac59

1 solution

A Former Brilliant Member
Oct 12, 2015

The probability of hitting the target by shooting 20 bullets is $\frac{8}{9}$ . So the probability of not hitting the target by shooting 20 bullets is $\frac{1}{9}$ . Let q be the probability of each bullet not hitting the target. So $q^{20}$ = $\frac{1}{9}$ . The probability of 10 bullets not hitting the target is $q^{10}$ = $\frac{1}{3}$ . The probability of 10 bullets hitting the target is 1- $\frac{1}{3}$ = $\frac{2}{3}$

Nice job! An equivalent way to frame your solution is that "not hitting any of 20" is "not hitting any of 10 twice". In other words, if $p$ is the probability of hitting any of 10, then $1-\frac{8}{9} = (1-p)^2 \quad \Rightarrow \quad (1-p)^2 = \frac{1}{9} \quad \Rightarrow \quad 1-p = \frac{1}{3} \quad \Rightarrow \quad p=\frac{2}{3}.$

Eli Ross Staff - 5 years, 8 months ago

That's how I figured it out! (Not hitting any of 20 is not hitting any of 10 twice)

Ian McKay - 5 years, 8 months ago

Hey can you help me with this puzzle " Find the next number? 1, 4, 9, 22, 45, 27, 55, 58, 99 __"

Vamshi Krishna Kandula - 5 years, 7 months ago

Shoot The Target

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