That is in all probability nearer to being suited to the arithmetic SE, however let’s undergo it.
Additionally, funnily sufficient so far as I can inform there is no such thing as a scoring choice for “Two Pairs” in Yahtzee as formally launched by Hasbro. In Yatzy or Yacht, that are frequent variants launched elsewhere, the values of the pairs should be distinct values, however the fifth die could also be a replica of one of many pairs (i.e. you may rating a Full Home as Two Pairs, however not 4 of a Variety). I might be engaged on that assumption right here.
Rolling all
For those who roll all 5 cube, then there are 6^5 = 7776 potential rolls.
To make Two Pair with an off-value fifth die, there are 6 methods to decide on the off-value, and 10 alternatives for the values of the pairs. The off-value die might be any of 5 decisions, after which there are 6 methods the 2 pairs might be positioned among the many different 4 cube. Therefore there are 6x10x5x6 = 1800 methods to roll Two Pair with an off-value.
As well as, to make a Full Home you’ve gotten 6 decisions of worth for the triple, and 5 decisions for the double (a Yatzy cannot be scored as Two Pair). There are 10 methods to decide on which cube kind the double, so there are 300 methods to roll a Full Home.
Therefore, there are 1800+300=2100 methods to roll Two Pair, and a likelihood of 2100/7776 = 27.0% that you’ll achieve this.
Retaining the pair
For those who simply preserve the pair of 3s, you are rolling 3 cube (of which there are 6^3 = 216 combos) and hoping for both:
- 3 of the identical worth, not equal to three.
- A pair not equal to three, and a price not equal to it (however probably equal to three).
For (1), there are 5 potential values for the triple, and for every of these there’s precisely one method to roll it (since 111 cannot be re-arranged to a special roll).
For (2), there are 5 potential values for the pair, and 5 potential values for the off-value (together with 3 however not the brand new pair), and there are 3 cube which might maintain the off-value, so there are 75 completely different rolls.
Therefore there are 80 rolls to get Two Pair, and a likelihood of 80/216 = 37.0%.
Retaining the pair + 1
On this case, you are rolling 2 cube (and there are 6^2 = 36 potential rolls), and to get your Two Pair you should roll a 5 on not less than one die, with the opposite worth being something you need. There are many methods to calculate this, however there are 11 methods for this to occur (e.g. 6 methods to roll 5 on the primary die, plus 6 methods to roll it on the second, however subtract one as a result of we have double-counted the roll of 55). So the likelihood is 11/36 = 30.5%
This is sensible, because you’re saving the “progress” of getting already rolled one of many wanted pairs (simply as within the second choice), however you are additionally proscribing your self to solely permitting one potential worth of second pair present up.
