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AMC 8 Mock questions

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Re: AMC 8 Mock questions
by John Lensmire - Monday, December 4, 2023, 11:39 AM

Let me again give some hints to help.

1st Question (probability): The hardest part of this problem is determining the probability that a dart (on a single throw) lands in the various regions. The probability is proportional to the area of each region. To start, notice that the inner circle has an area of $\pi 3^2 = 9\pi$, the full circle has area $\pi\cdot 6^2 = 36\pi$, and therefore the outer ring has probability $36\pi - 9\pi = 27\pi$.

Therefore, each of the inner regions (with points 1, 2, and 2) have probability $\dfrac{9\pi \div 3}{36\pi} = \dfrac{1}{12}$ and each of the outer regions (with points 2, 1, and 1) have probability $\dfrac{27\pi\div 3}{36\pi} = \dfrac{1}{4}$.

Some hints to continue from here: (i) What is the probability a single throw is 1 (odd)? What is the probability a single throw is 2 (even)? How can the sum of the two throws be odd?

2nd Question (arithmetic & logic): Actually I think the problem does say that each region can only be hit once, because it says "Each throw hits the target in a region with a different value." Note: Even though both problems are about darts, they really are not very similar to each other, since the second one doesn't really have any probability involved.

Hope this helps!