Difference between revisions of "2019 AIME II Problems/Problem 2"
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| + | ==Problem== | ||
Lily pads <math>1,2,3,\ldots</math> lie in a row on a pond. A frog makes a sequence of jumps starting on pad <math>1</math>. From any pad <math>k</math> the frog jumps to either pad <math>k+1</math> or pad <math>k+2</math> chosen randomly with probability <math>\tfrac12</math> and independently of other jumps. The probability that the frog visits pad <math>7</math> is <math>\tfrac pq</math>, where <math>p</math> and <math>q</math> are relatively prime positive integers. Find <math>p+q</math>. | Lily pads <math>1,2,3,\ldots</math> lie in a row on a pond. A frog makes a sequence of jumps starting on pad <math>1</math>. From any pad <math>k</math> the frog jumps to either pad <math>k+1</math> or pad <math>k+2</math> chosen randomly with probability <math>\tfrac12</math> and independently of other jumps. The probability that the frog visits pad <math>7</math> is <math>\tfrac pq</math>, where <math>p</math> and <math>q</math> are relatively prime positive integers. Find <math>p+q</math>. | ||
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| + | ==Solution== | ||
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| + | ==See Also== | ||
| + | {{AIME box|year=2019|n=II|num-b=1|num-a=3}} | ||
| + | {{MAA Notice}} | ||
Revision as of 16:56, 22 March 2019
Problem
Lily pads 
lie in a row on a pond. A frog makes a sequence of jumps starting on pad 
. From any pad 
the frog jumps to either pad 
or pad 
chosen randomly with probability 
and independently of other jumps. The probability that the frog visits pad 
is 
, where 
and 
are relatively prime positive integers. Find 
.
Solution
See Also
| 2019 AIME II (Problems • Answer Key • Resources) | ||
| Preceded by Problem 1 |
Followed by Problem 3 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
