Question
(Difficulty: 6/10)
In a country, each of the citizens has a 6-digit ID number. For example, 077493 is a valid ID number. Now the country holds a lucky draw: citizens having all the three numbers '6', '8' and '9' in the ID number can have the chance of slapping the Chief Executive. For example, 066189 can slap the Chief Executive but 691674 cannot.
What is the probability that a citizen can slap the Chief Executive?
Assume:
- There are not more than 1,000,000 people in this country
- Each citizen has a unique ID number
- ID number are randomly generated for each citizen, i.e. the first man in this country may have an ID 729412
Solution
The probability we want to find is Pr(has 6 AND has 8 AND has 9)
Finding this may not be easy, so let's consider its opposite: Pr(no 6 OR no 8 OR no 9)
Let's consider the Venn Diagram below, which is a great tool to solve probability problem.
Venn Diagram
Let's consider three events: no 6, no 8 and no 9. Can we break down the event of {no 6 OR no 8 OR no 9} into {no 6} + {no 8} + {no 9}? No! Because in this way we would have double-counted some events, such as {no 6 AND no 8}.
Now let's draw a Venn Diagram, which helps us to visualize the events.
We now have 3 circles, which represents the three events: {no 6}, {no 8}, {no 9}. However these three circles have some intersection areas, like D, E, F and G.
Here is a list of the meaning of each collection of areas:
- A+D+F+G: {no 6}
- C+E+F+G: {no 8}
- B+D+E+G: {no 9}
- F+G: {no 6 AND no 8}
- D+G: {no 6 AND no 9}
- E+G: {no 8 AND no 9}
- G: {no 6 AND no 8 AND no 9}
- A+B+C+D+E+F+G: {no 6 OR no 8 OR no 9}
- H: {has 6 AND has 8 AND has 9} What we want!
Therefore, what we need to find is Pr(H), which can be done by finding the area of (A+B+C+D+E+F+G) first.
Here is our final result (you can try to verify it by considering the areas that these events represent):
Pr(no 6 OR no 8 OR no 9) = Pr(no 6) + Pr(no 8) + Pr(no 9) - Pr(no 6 AND no 8) - Pr(no 6 AND no 9) - Pr(no 8 AND no 9) + Pr(no 6 AND no 8 AND no 9)
This is also known as the inclusion-exclusion principle.
Calculations
Pr(no 6) = Pr(no 8) = Pr(no 9) = 0.9^6
Pr(no 6 AND no 8) = Pr(no 6 AND no 9) = Pr(no 8 AND no 9) = 0.8^6
Pr(no 6 AND no 8 AND no 9) = 0.7^6
Therefore Pr(no 6 OR no 8 OR no 9) = 0.92554
Finally we have the required answer = 1 - 0.92554 = 0.07446
Congrats to @kelkoo !
Previous posts
Math challenge #1 - Find the remainder!
Math challenge #2 - Find the value of these recurring square roots
Math challenge #3 - Find its length!
Interested in Math puzzles? Follow @kenchung, upvote and resteem! :)
My answer is 0.07446 :)
Great! Thats the correct answer. Let's not disclose the method first so that others can have a chance to think ;)
What a wonderful challenge. I must admit my math skills are a bit rusty though I was left drooling haha.
-- A little about myself --
I am a photographer, writer, and videographer (Snapchatter really @faceofbear 😉)
I currently live in Barcelona, Spain. Each day, I seek out the beauty around me. I have traveled the world on a spiritual journey to find a deeper connection and understanding. I hope to bring to the Steem community quality content and stories about my discoveries. Feel free to take a look at my most recent journey.
https://steemit.com/writing/@faceofbear/66dtff-angkor-wat-flirting-with-ancient-wisdoms-part-1-of-2
Have a lovely day 👻
Thanks for stopping by! You can participate in the next math challenge so as to practise your maths skills :)
Just read your post and really like it! your photos are so beautiful. looking forward to your next post! upvoted and followed :)
This is so awesome to see. Math is beautiful, and I'm so glad to see someone cherish it like this!
Have you ever thought about using your math challenges and turning them into programming challenges? The two fields are very similar as they share the logical, problem-solving core knowledge one would need. Let me know, I'd love to give you a hand with that.
sure, that's a really good idea! I have some basic programming knowledge and I love it too! Just like this question, actually it can be solved by programming as well.
Would love to hear your opinion on making this into programming challenges as well!
Awesome! I'll shoot you a message on Steemit.chat later today for a brainstorming session. :)
sure! :)
Just to clarify, by "both" do you mean "all three", or perhaps "at least two from those three"? Thanks
In fact I meant 'all three', thanks for asking! I should have used the word "all". Amended.
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very nice puzzle. Very difficult.
Yup this time the difficulty is 6 out of 10 ;) Next time I may lower the difficulty so that more people can participate haha