A probabilistic approach towards counting

Imagine you have a keyboard that only consists of 3 keys; backspace, A, and B. You decide to roll a regular fair dice to determine which key is going to be chosen each time and follow the rules below:
1- If the number on the dice is 3 or 6, you choose backspace.
2- If you see number 2 or 5, you type B.
3- If numbers 1 or 4 appear then you'll press A.
How many steps on average does it require to type out the string "ABBB"?
P.S. would it make any difference if the desired string is a substring as in "AAABABBBA"?

#Combinatorics

Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

According to my calculation, answer should be 63. My solving strategy was to use state diagram. States are: "/, A, AB, ABB, ABBB" . Then, you assign a probability to each state transition (for example, once you are in state "AB", there's 1/3 chance to go to state "ABB" and 2/3 chance to go to state "A"). Based on that, you write out equations for expected value of steps starting from each state and, by going from back to the front, you solve for expected value of steps starting from the "/" state.

Uros Stojkovic - 1 year, 10 months ago

Can you provide a detailed proof? How will the expected value of each state relates to getting average number of attempts to get there?

shivam jalotra - 1 year, 7 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$