How many times must we roll a single die in order to get the same score:
a) at least twice?
b) at least three times?
c) at least n times? for n >= 4
we still have 6 (n) holes, i thought we just need another pigeons, total of 8 pigeons
JSuarez said:No, for all questions you have just six "holes" (the faces), but the number of "pigeons" (each pigeon is the score at each throw) increases.
So for the second (and third) questions, just apply the same reasoning that you applied on the first. You should be able to see that 8 doesn't work; just look at the following eight (possible) scores: 1 4 3 2 4 2 1 6
So, how many throws do you need for the same score to appear at least three times?
The Pigeonhole Principle, also known as the Dirichlet's box principle, states that if n pigeons are placed into m boxes, with n > m, then at least one box must contain more than one pigeon.
When rolling a die, there are 6 possible outcomes (1, 2, 3, 4, 5, 6) and only 5 possible pigeonholes (each side of the die). Therefore, at least one number must appear on multiple sides of the die, fulfilling the conditions of the Pigeonhole Principle.
The Pigeonhole Principle is often used in probability to prove the existence of certain outcomes. It can be applied to show that in a set of random events, there is a high likelihood of at least one outcome occurring more than once.
The Pigeonhole Principle challenges the idea of true randomness by showing that in a set of possible outcomes, some outcomes are more likely to occur multiple times than others. This suggests that true randomness may not exist in certain scenarios.
Yes, the Pigeonhole Principle can be applied to various real-life scenarios, such as organizing items into categories, scheduling appointments, and even in the field of computer science. It is a useful tool for understanding patterns and predicting outcomes.