# An interesting probability!

Let describe a process $A$ as follows:

Suppose you have a bag, and it have a ball which either have $\red{red}$ or $\blue{blue}$ color (you don't know what color it have). You put a $\red{red}$ ball inside that bag and pick a ball at random from the bag, let the color of the ball you got be called the output of the process and the ball left inside the bag as the main ball of the process.

Now suppose you do process $A$ ''$n$'' times and every time you get $\red{red}$ ball as output then the probability that the main ball of the whole process is $\blue{blue}$ is $\dfrac{100}{2^n+1}\%$

Proof :

Let us analyze each scenario one by one

First scenario (The ball in the start was of $\blue{blue}$ color)

In this case there is only one and only one output coming every time , that is the red ball you put inside the bag in starting is the same ball you are getting as output each time you does the process

Second scenario (The ball in the start was of $\blue{blue}$ color)

In this scenario, each time you will be getting either of the two $red$ ball as an output.

So in total how many different ways can you pick ball from the bag and then put it in the bag and then again pick a ball, and does this $n$ times? You can prove it yourself, is $2^n$

So total number of possible events $=2^n+1$

Number of possible events in which there is $\blue{blue}$ ball left$=1$

$\therefore$ probability that the ball left inside the bag after applying the process $n$ times $=\dfrac{1}{2^n+1}$

For percentage multiply it by $100$ getting the probability percentage that the main ball after applying the process $n$ times $=\dfrac{100}{2^n+1}\%$

Note :

• Inspiration

• Meaning of doing process $A$ ''$n$'' times : It means that you put the red ball inside the bag and then pick a ball randomly, after that you put the ball back into the bag and then again pick a ball randomly and does this continuously ''$n$'' times and after doing it $n_{th}$ time you don't put the output ball back in the bag this time, and the other ball left in the bag is the main ball of the whole process Note by Zakir Husain
11 months, 3 weeks 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.
• 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.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. 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 1paragraph 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"
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} $\boxed{123}$

## Comments

Sort by:

Top Newest

@Jeff Giff- See if you find this interesting enough to add in $\large{RadMaths}$

- 11 months, 3 weeks ago

Log in to reply

Interesting @Zakir Husain

- 11 months, 3 weeks ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...
{"ft_lang_label":"__label__en","ft_lang_prob":0.9161629,"math_prob":0.9827943,"size":2666,"snap":"2021-21-2021-25","text_gpt3_token_len":625,"char_repetition_ratio":0.15401953,"word_repetition_ratio":0.06762295,"special_character_ratio":0.23518379,"punctuation_ratio":0.07777778,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9964852,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-06-24T01:15:43Z\",\"WARC-Record-ID\":\"<urn:uuid:f99518d7-44c2-4a3e-a6a8-4058e0d42425>\",\"Content-Length\":\"77765\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:ac7d8f7f-9c9a-4822-bb89-9693f49e4b0e>\",\"WARC-Concurrent-To\":\"<urn:uuid:5f32255a-5994-45d4-b03c-eac0977d8f87>\",\"WARC-IP-Address\":\"104.20.35.242\",\"WARC-Target-URI\":\"https://brilliant.org/discussions/thread/an-interesting-probability/\",\"WARC-Payload-Digest\":\"sha1:ESGWD2GBU32QW2X3CI4GNRDOW46ZRUXW\",\"WARC-Block-Digest\":\"sha1:RL6FHYXJS2OXUMX4VQXPSMY2H6JQZQ2F\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-25/CC-MAIN-2021-25_segments_1623488544264.91_warc_CC-MAIN-20210623225535-20210624015535-00032.warc.gz\"}"}