Two identical indistinguishable symmetric dices once and record the result of this tossing (i.e. "on one dice we obtained 1 and on another we obtained 2").
What is the probability of event "on one die we obtained 1 and on another we obtained 2"? What is the probability of event "obtaining two 1's"? What kind of sample space is considered in this problem: sample space with equal probabilities of outcomes or sample space with non-equal probabilities of outcomes?
The following approach i have used for the above problem statement
Here two identical indistinguishable symmetric dices are used hence the given experiment has fair die where each outcome is not equally likely as pairs for example (1,2)(2,1) are same so it will have probabilities 1/18 instead of 1/36 and thus the sample space will have non equal probabilities of the outcomes. The sample space of this experiment will have all the possible outcomes recorded in the dice but since the dice are identical and indistinguishable so pairs like (1,2) ,(2,1) only one would be counted so removing such pairs which are 15 from the list of all possible outcomes we get the cardinality of the sample space as 36-15=21
Let A be an event as "on one dice we obtained 1 and on another we obtained 2".
thus P(A)=1/21 as number of ways to get 1 and 2 on both die is 1 and number of all possible outcomes is 21
Let B be an event as "we obtained two 1's".
thus P(B)=1/21 as number of ways to get two 1's on both die is 1 and number of all possible outcomes is 21
Can you please let me know if the above approach is correct for the above problem statement?
Best Answer
Close but wrong.
The statement that the dice are indistinguishable is very misleading.
The only way to attack such a problem is through intuition.
Imagine instead that one die is red and one die is green.
Then, it becomes immediately obvious that there are
2 ways (out of 36) of rolling a 1,2
either green=1, red=2, or vice versa.
Then, there is only 1 way out of 36 of rolling 1,1
green = 1, red=1.
The whole key to the problem is to understand, that if the red die is replaced by a green die, so that the dice are now indistinguishable, the chances of each roll can not have been affected.
This understanding only comes through intuition. And the intuition only comes by attacking probability/combinatorics problems. So you were trapped in a catch-22 situation.