Lillie Pittman

2022-07-17

The probability of the dart hitting the smaller ring is?
A dart is randomly thrown at a circular board on which two concentric rings of radii R and 2R having the same width (width much less than R) are marked. The probability of the dart hitting the smaller ring is:
(1) Twice the probability that it hits the larger ring.
(2) Half of the probability that it hits the larger ring.
(3) Four times the probability that it hits the larger ring.
(4) One-fourth the probability that it hits the larger ring.

emerhelienapj

Expert

Step 1
Let's say the common width was W. Then the area occupied by a ring of radius r would be $\pi \left(r+W{\right)}^{2}-\pi {r}^{2}=2\pi rW+\pi {W}^{2}$
Thus the probability that the dart will land in that ring is $\frac{2\pi rW+\pi {W}^{2}}{\pi {S}^{2}}=\frac{2rW}{{S}^{2}}+\frac{{W}^{2}}{{S}^{2}}\approx \frac{2rW}{{S}^{2}}$ where we have used the fact that W is much smaller than S.
Step 2
It follows that the ratio you want is approximately $\frac{2RW}{{S}^{2}}/\frac{4RW}{{S}^{2}}=\frac{1}{2}$

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