Hi Guys, long time since I last posted, but I still regularly read the forum. I have stumbled on an irresistible probability puzzle, which I could not actually solve analytically. So I wrote a little program for a brute force solution, counting the outcomes. Here is the little simple-sounding puzzle:

*In a standard 3-dice roll, what is the probability of any one or more pairs adding up to 10 or more?*

Here are the qualifying rolls, according to my .js script:

146 155 156 164 165 166 246 255 256 264 265 266 346 355 356 364 365 366 416 426 436 446 455 456 461 462 463 464 465 466 515 516 525 526 535 536 545 546 551 552 553 554 555 556 561 562 563 564 565 566 614 615 616 624 625 626 634 635 636 641 642 643 644 645 646 651 652 653 654 655 656 661 662 663 664 665 666

3 dice, count = 77 out of (6^{3} = 216), giving a probably of 35.65%

So what does the combination/permutation formula look like, in terms of N dice?

As a check of the formula, the same script tells me for a 4-dice roll, the result is:

4 dice, count = 676 out of (6^{4} = 1296), giving a probably of 52.16%

I honestly could not find a formula for the general case!

Any ideas?

## Comments rated to be "almost" Good Answers:

Re: Dice Probability Puzzle" by Rixter on 07/21/2020 12:51 PM (score 1)Re: Dice Probability Puzzle" by Rixter on 07/20/2020 6:45 PM (score 1)Re: Dice Probability Puzzle" by Rixter on 07/19/2020 6:12 PM (score 1)