Reflections on Probability Problem#2

Here's another problem that I actually got correct, but just to compare my reasoning with the "view result" I clicked there and the explanation is so weird. Here's the problem and it's solution follows:

Two people, Alice and Bill, each roll a fair 20-sided die. What is the probability that Alice's roll is higher than Bill's roll?

Solution: The probability that they roll the same number is $\frac{1}{20},$ so the probability that they do not roll the same number is $\frac{19}{20}.$ By symmetry, if they do not roll the same number, each of Alice and Bill is equally likely to have rolled the higher number. The answer is thus $\frac{1}{2} \cdot \frac{19}{20} = \frac{19}{40}.$

What I was thinking: Bill either rolls 1, or 2, or,...., or 19. Then Alice beats Bill if she gets any of the next 19 numbers, or the next 18 numbers, or,....,or the last number 20. Then using a formula, Alice can beat Bill in $\frac{19(19+1)}{2}$ ways which give same result.

My question: Alice and Bill having equally likely chances to roll higher means that we're like flipping a coin? How will I be able to spot this strategy in other problems? If I were to write this in words, does this mean that we are calculating P(Alice wins) = P(They do not roll same and Bill loses) = P(They do not roll same)P(Bill loses) = P(They do not roll same and Alice wins)?

Reflections: I guess I do understand the technique. My previous paragraph kind of Illustrates what is meant by "symmetry" and equal likelihood. So I'm making this note more like a reflexion kind of thing than a question.

Note by Jay B
3 years, 5 months ago

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.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

• bulleted
• list

1. numbered
2. list

1. numbered
2. 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

> 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"
MathAppears 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} $\boxed{123}$

Sort by:

I am also confused by this problem. Can somebody explain to me how the probability they roll the same is 1/20 shouldn't it be 1/20 * 1/20

- 3 years ago

The number of eyes of the first roll doesn't really matter. Let's say Alice rolls a 2, the chance that Bill rolls a 2 is 1/20 (But Alice can also roll a 15 for example, than Bill has to roll a 15 as well, which has a probability of 1/20).

- 2 years, 4 months ago

Thank you man for the explaination

- 1 year, 5 months ago