The Question: You have 20 standard dice (6 sides,numbered from 1-6). You roll th

Question

The Question:
You have 20 standard dice (6 sides,numbered from 1-6).
You roll the dice and collect all thedice which roll an even number.
You then roll the dice you collected,and this time collect all the dice which roll a 1.
You roll the new collection ofdice.
What is the probability that none of thedice on your final roll display a 1 or a 2 (if you have acollection of zero dice in your initial or final collection, thenyou will automatically have none of the dice on your finalroll display a 1 or a 2).
I know there's a formula for finding it out quicker than howI'm doing it, this is what I have so far, but it seems like I'll bethere for pages trying to do it this way:
In the first roll there is 6^20 combinations. There is 3^20combinations that will give an initial collection of 0. There is3^20 combinations that will give an initial collection of 20, andthen 5^20 of the 6^20 combinations that give a final collection of0. There are 3^20 x 20 combinations of an initial collection of 19,and then 5^19 of the 6^19 combinations will give a final collectionof 0. There is 3^20 x 20 combinations that give an initialcollection of 1, and then 5/6 combinations that give us a finalcollection of 0.
Please help me! The Question:
You have 20 standard dice (6 sides,numbered from 1-6).
You roll the dice and collect all thedice which roll an even number.
You then roll the dice you collected,and this time collect all the dice which roll a 1.
You roll the new collection ofdice.
What is the probability that none of thedice on your final roll display a 1 or a 2 (if you have acollection of zero dice in your initial or final collection, thenyou will automatically have none of the dice on your finalroll display a 1 or a 2).
I know there's a formula for finding it out quicker than howI'm doing it, this is what I have so far, but it seems like I'll bethere for pages trying to do it this way:
In the first roll there is 6^20 combinations. There is 3^20combinations that will give an initial collection of 0. There is3^20 combinations that will give an initial collection of 20, andthen 5^20 of the 6^20 combinations that give a final collection of0. There are 3^20 x 20 combinations of an initial collection of 19,and then 5^19 of the 6^19 combinations will give a final collectionof 0. There is 3^20 x 20 combinations that give an initialcollection of 1, and then 5/6 combinations that give us a finalcollection of 0.
Please help me!

Explanation / Answer

I just corrected my mistake in red. I had forgotten to includethe event that X = 0. Let's hope I'm right this time. Best, Mike

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