4th January 2021, 1:58 PM
Hello all!
So, I don't know about the rest of you but I really like math. . . So every so often I will post a new puzzle that has a very interesting solution behind it. If people seem interested that I will keep on posting puzzles. There are not prizes besides the knowledge about our universe!
Puzzle 1:
Consider the following situation: Rolling a 5-of-a-kind with a total of five, six sided dice. each die has an equally fair chance of rolling a number between 1 through 6. The odds of rolling a 5-of-a-kind is equal to 1/1296 (or 6 desired outcomes to 7776 total outcomes).
While attempting to roll a 5-of-a-kind is unlikely it is not impossible, so attempting this over and over again will increase your chances of achieving this desired result. Now, lets suppose you took 1296 attempts to get a 5-of-a-kind. Since the odds are 1/1296 it is likely that you would have seen this result at least once. The first question is: What is the probability that you would have seen a 5-of-a-kind at least once in 1296 rolls.
This is a very interesting question because even with slim odds and a very large number of attempts you are still not guaranteed to see the desired result within the 1296 tries. This question is also interesting because it a question that considers the "Probability of Probabilities."
Question 2: Once you work out the solution to question 1, determine how many rolls will give you a 50% chance of seeing a 5-of-a-kind at least once. A 75% chance? a 99% chance?
Question 3: Lets consider you have a 1 in 'n' chance of a desired outcome (where n is an arbitrary whole number) and you had an 'n' amount of attempts to achieve the desired result. What happens as n gets really large (That would imply that the odds of a given outcome is getting smaller as 1/n goes to zero, but also, you are giving that event more chances at showing up with that larger number."
Good luck!
So, I don't know about the rest of you but I really like math. . . So every so often I will post a new puzzle that has a very interesting solution behind it. If people seem interested that I will keep on posting puzzles. There are not prizes besides the knowledge about our universe!
Puzzle 1:
Consider the following situation: Rolling a 5-of-a-kind with a total of five, six sided dice. each die has an equally fair chance of rolling a number between 1 through 6. The odds of rolling a 5-of-a-kind is equal to 1/1296 (or 6 desired outcomes to 7776 total outcomes).
While attempting to roll a 5-of-a-kind is unlikely it is not impossible, so attempting this over and over again will increase your chances of achieving this desired result. Now, lets suppose you took 1296 attempts to get a 5-of-a-kind. Since the odds are 1/1296 it is likely that you would have seen this result at least once. The first question is: What is the probability that you would have seen a 5-of-a-kind at least once in 1296 rolls.
This is a very interesting question because even with slim odds and a very large number of attempts you are still not guaranteed to see the desired result within the 1296 tries. This question is also interesting because it a question that considers the "Probability of Probabilities."
Question 2: Once you work out the solution to question 1, determine how many rolls will give you a 50% chance of seeing a 5-of-a-kind at least once. A 75% chance? a 99% chance?
Question 3: Lets consider you have a 1 in 'n' chance of a desired outcome (where n is an arbitrary whole number) and you had an 'n' amount of attempts to achieve the desired result. What happens as n gets really large (That would imply that the odds of a given outcome is getting smaller as 1/n goes to zero, but also, you are giving that event more chances at showing up with that larger number."
Good luck!
