Difference between revisions of "2019 AMC 12B Problems/Problem 16"

Revision as of 20:58, 19 February 2019

Problem

There are lily pads in a row numbered $0$ to $11$ , in that order. There are predators on lily pads $3$ and $6$ , and a morsel of food on lily pad $10$ . Fiona the frog starts on pad $0$ , and from any given lily pad, has a $\frac{1}{2}$ chance to hop to the next pad, and an equal chance to jump $2$ pads. What is the probability that Fiona reaches pad $10$ without landing on either pad $3$ or pad $6$ ?

$\textbf{(A) } \frac{15}{256} \qquad \textbf{(B) } \frac{1}{16} \qquad \textbf{(C) } \frac{15}{128}\qquad \textbf{(D) } \frac{1}{8} \qquad \textbf{(E) } \frac14$

Solution 1

Firstly, notice that if Fiona jumps over the predator on pad $3$ , she must on pad $4$ . Similarly, she must land on $7$ if she makes it past $6$ . Thus, we can split the problem into $3$ smaller sub-problems, separately finding the probability Fiona skips $3$ , the probability she skips $6$ (starting at $4$ ) and the probability she doesn't skip $10$ (starting at $7$ ). Notice that by symmetry, the last of these three sub-prolems is the complement of the first sub-problem, so the probability will be $1 - \text{the probability obtained in the first sub-problem}$ .

In the analysis below, we call the larger jump a $2$ -jump, and the smaller a $1$ -jump.

For the first sub-problem, consider Fiona's options. She can either go $1$ -jump, $1$ -jump, $2$ -jump, with probability $\frac{1}{8}$ , or she can go $2$ -jump, $2$ -jump, with probability $\frac{1}{4}$ . These are the only two options, so they together make the answer $\frac{1}{8}+\frac{1}{4}=\frac{3}{8}$ . We now also know the answer to the last sub-problem is $1-\frac{3}{8}=\frac{5}{8}$ .

For the second sub-problem, Fiona must go $1$ -jump, $2$ -jump, with probability $\frac{1}{4}$ , since any other option would result in her death to a predator.

Thus, since the three sub-problems are independent, the final answer is $\frac{3}{8} \cdot \frac{1}{4} \cdot \frac{5}{8} = \boxed{\textbf{(A) }\frac{15}{256}}$ .

Solution 2

Observe that since Fiona can only jump at most $2$ places per move, and still wishes to avoid pads $3$ and $6$ , she must also land on numbers $2$ , $4$ , $5$ , and $7$ .

There are two ways to reach lily pad $2$ , namely $1$ -jump, $1$ -jump, with probability $\frac{1}{4}$ , or just a $2$ -jump, with probability $\frac{1}{2}$ . The total is thus $\frac{1}{4} + \frac{1}{2} = \frac{3}{4}$ . Fiona must now make a $2$ -jump to lily pad $4$ , again with probability $\frac{1}{2}$ , giving $\frac{3}{4} \cdot \frac{1}{2} = \frac{3}{8}$ .

Similarly, Fiona must now make a $1$ -jump to reach lily pad $5$ , again with probability $\frac{1}{2}$ , giving $\frac{3}{8} \cdot {1}{2} = \frac{3}{16}$ . Then she must make a $2$ -jump to reach lily pad $7$ , with probability $\frac{1}{2}$ , yielding $\frac{3}{16} \cdot \frac{1}{2} = \frac{3}{32}$ .

Finally, to reach lily pad $10$ , Fiona has a few options - she can make $3$ consecutive $1$ -jumps, with probability $\frac{1}{8}$ , or $1$ -jump, $2$ -jump, with probability $\frac{1}{4}$ , or $2$ -jump, $1$ -jump, again with probability $\frac{1}{4}$ . The final answer is thus $\frac{3}{32} \cdot \left(\frac{1}{8} + \frac{1}{4} + \frac{1}{4}\right) = \frac{3}{32} \cdot \frac{5}{8} = \boxed{\textbf{(A) } \frac{15}{256}}$ .

Solution 3 (recursion)

Let $p_n$ be the probability of landing on lily pad $n$ . Observe that if there are no restrictions, we would have $p_n = \frac{1}{2} \cdot p_{n-1} + \frac{1}{2} \cdot p_{n-2}$

This is because, given that Fiona is at lily pad $n-2$ , there is a $\frac{1}{2}$ probability that she will make a $2$ -jump to reach lily pad $n$ , and the same applies for a $1$ -jump to reach lily pad $n-1$ . We will now compute the values of $p_n$ recursively, but we will skip over $3$ and $6$ . That is, we will not consider any jumps from lily pads $3$ or $6$ when considering the probabilities. We obtain the following chart, where an X represents an unused/uncomputed value:

$[asy] unitsize(40); string[] vals = {"1", "$1/2$", "$3/4$", "X", "$3/8$", "$3/16$", "X", "$3/32$", "$3/64$", "$9/128$", "$15/256$", "X"}; for(int i =0; i<= 11; ++i) { draw((i,0)--(i+1,0)--(i+1,1)--(i,1)--cycle); label((string) i, (i+0.5,0), S); label(vals[i], (i+0.5, 0.5)); } [/asy]$

Hence the answer is $\boxed{\textbf{(A) } \frac{15}{256}}$ .

@@ Line 11: / Line 11: @@
 For the first sub-problem, consider Fiona's options. She can either go <math>1</math>-jump, <math>1</math>-jump, <math>2</math>-jump, with probability <math>\frac{1}{8}</math>, or she can go <math>2</math>-jump, <math>2</math>-jump, with probability <math>\frac{1}{4}</math>. These are the only two options, so they together make the answer <math>\frac{1}{8}+\frac{1}{4}=\frac{3}{8}</math>. We now also know the answer to the last sub-problem is <math>1-\frac{3}{8}=\frac{5}{8}</math>.
-For the second sub-problem, Fiona <math>\textit{must}</math> go <math>1</math>-jump, <math>2</math>-jump, with probability <math>\frac{1}{4}</math>, since any other option would result in her death to a predator.
+For the second sub-problem, Fiona ''must'' go <math>1</math>-jump, <math>2</math>-jump, with probability <math>\frac{1}{4}</math>, since any other option would result in her death to a predator.
 Thus, since the three sub-problems are independent, the final answer is <math>\frac{3}{8} \cdot \frac{1}{4} \cdot \frac{5}{8} = \boxed{\textbf{(A) }\frac{15}{256}}</math>.

2019 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 15	Followed by Problem 17
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