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Theoretical Probability of Rolling a Yahtzee p = 1/6 = chance of getting a specific number q = 1p = 5/6 = chances of failing. Chances of getting 1,2,3,4,5 of a specific number after 1 throw of 5 dice * 0 match q^{5} ==> 42% * 1 match 5pq^{4} ==> 42% * 2 matches 10p^2q^{3} ==> 16% * 3 matches 10p^3q^{2} ==> 3.2% * 4 matches 5p^4q ==> 0.32% * 5 matches p^{5} ==> 0.013% the above is the expansion of (p+q)^{5} Chances of getting 1,2,3,4,5 of a specific number after 2 throws of 5 dice * 0 match q^{10} ==> 16% * 1 match 5p(1+q)q^{8} ==> 36% * 2 matches 10p^{2}(1+q)^2q^{6} ==> 31% * 3 matches 10p^{3}(1+q)^3q^{4} ==> 14% * 4 matches 5p^{4}(1+q)^4q^{2} ==> 3.0% * 5 matches p^{5}(1+q)^{5} ==> 0.27% the above is the expansion of (p(1+q) + q^{2})^{5} Chances of getting 1,2,3,4,5 of a specific number after 3 throws. * 0 match q^{15} ==> 6.5% * 1 match 5p(1+q+q^{2})q^{12} ==> 24% * 2 matches 10p^{2}(1+q+q^{2})^2q^{9} ==> 34% * 3 matches 10p^{3}(1+q+q^{2})^3q^{6} ==> 25% * 4 matches 5p^{4}(1+q+q^{2})^4q^{3} ==> 9.1% * 5 matches p^{5}(1+q+q^{2})^{5} ==> 1.3% the above is the expansion of (p(1+q+q^{2}) + q^{3})^{5} Any yahtzee is very difficult, as you can change direction on the 2nd throw (e.g. 1st throw is 1,1,3,4,5 then you throw 1,1,6,6,6, you now go for 6s) but your web site has a good explaination of that (though, my calculations suggest 4.6% is the chances of getting any Yahtzee, and not 3.7%) Any Yahtzee after 1 throw 0.077% Any Yahtzee after 2 throws 1.3% Any Yahtzee after 3 throws 4.6% Adding it all up we have a 4.6% chance of a Yahtzee which is roughly 1 in 22. So, for every 27 tries you should expect to get one Yahtzee over the long run. In real play only 10 goes target fivealike (excluding full house and the two straights) so you should expect one yahtzee per 2.2 games.
