Online Course Discussion Forum

2019 amc12b #16

 
 
Picture of Jialin Wang
2019 amc12b #16
by Jialin Wang - Sunday, February 6, 2022, 7:35 PM
 

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 $6$?


for this one, i separated this into 3 parts: before pad 3, between 3 and 6, after 6. And there are 2 ways for the frog to jump: 1 pad or 2 pads.

I figured out that for before pad 3 there are 2 ways: 1-1-2 or 2-2 so it's 1/8+1/4=3/8.

For between 3 and 6 there is only 1 way:1-2,  so it's 1/4

For the after 6 part , the answer says it should be 5/8 but I don't understand how this works. Is it supposed to be solved by stars and bars or something else?

 
Picture of John Lensmire
Re: 2019 amc12b #16
by John Lensmire - Monday, February 7, 2022, 9:49 AM
 

Good job on the earlier parts!

In fact, you want to use the same method for the last part. After jumping over pad 6, the frog is at pad 7 and must get to pad 10. Remember, however, that since there is a pad 11, the frog can still jump over the food on pad 10, which we need to avoid.

Hence, the possible jumps are 1-1-1, 1-2, or 2-1, with probability 1/8+1/4+1/4 = 5/8 for the last part. Then we just need to multiply all the parts together for the final answer.

Note: The method above works great, as the different cases are not too bad. I also, however, would recommend trying a recursive approach to the problem. If $P_N$ is the probability the frog gets to pad $N$, we know $P_0 = 1$ and $P_3=P_6=0$. How can you write $P_N$ in terms of the previous pads?