2023 AMC 10B Problems/Problem 19

froggo

Solution 1

WLOG, we assume Sonya jumps $0.5$ units every time, since that is her expected value.

If Sonya is within $0.5$ blocks of an edge, she can jump off the board. Let us examine the region that is at most $0.5$ blocks from exactly one edge.

$[asy] import graph; Label f; xaxis(0,6,Ticks(f, 6.0, 0.5)); yaxis(0,6,Ticks(f, 6.0, 0.5)); draw((0,0)--(6,0)--(6,6)--(0,6)--cycle); filldraw((0,0.5)--(0.5,0.5)--(0.5,5.5)--(0,5.5)--cycle,gray); filldraw((0.5,0)--(0.5,0.5)--(5.5,0.5)--(5.5,0)--cycle,gray); filldraw((6,0.5)--(5.5,0.5)--(5.5,5.5)--(6,5.5)--cycle,gray); filldraw((0.5,6)--(0.5,5.5)--(5.5,5.5)--(5.5,6)--cycle,gray); [/asy]$

If Sonya starts in this region, she has a $\dfrac14$ chance of landing outside (there's exactly one direction she can hop to get out). The total area of this region is $4\cdot0.5\cdot5=10.$ For this region, Sonya has a $\dfrac14$ chance, so we multiply $10$ by $\dfrac14$ to get $2.5.$

If Sonya is in one of the corner squares, she can go two directions to get out, so she has a $\dfrac24=\dfrac12$ chance to get out. The total area is $0.5\cdot0.5\cdot4=1$ , so this region yields $\dfrac12\cdot1=\dfrac12.$

Adding the two, we get $3$ . This is out of $36$ square units of area, so our answer is thus $\dfrac1{12}.$

~Technodoggo

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2023 AMC 10B Problems/Problem 19

Solution 1