Yesterday I had the situation of being at 250 and having to roll a single die. Given a choice one would never choose to do this, but Facebook rules require you roll till you get to 300. This relates to the problem of sitting at 250, but with the option of rolling either one die or two, i.e., you have two dice which scored, so you have the option of picking up a five and rolling it instead of taking the 50 points. I decided to figure the expected result rolling 1, 2 and 3 dice.
The possibilities are few enough that a brute force method is feasible. With one die, you will roll a one 1/6 of the time, for a total of 350, expected score here of 58.3. You will also roll a five 1/6 of the time, expected result here of 50. The rest of the time you farkle and lose the 250 you have, expected result of 0. Your expected score thus is 108.3.
Rolling two dice (when sitting at 200), there are 36 possible rolls. A 1-1 will come up 1/36 of the time, expected score of 400/36 or 11.1. A 5-5 will also come up 1/36 of the time, for expected score of 300/36, or 8.3. Similarly, 1-5 is 1/18, for 19.4, 1-X 8/36 for 66.7, and 5-X means you have to throw again, i.e., you are in the situation with one die. This means that 8/36 of the time you have an expoected result of 108.3, which figures to 24.1.
Add all these up and we get an expected result of 129.6. So, we have conclusively shown that when given a choice, one should pick up a five and roll two dice instead of just one.
When rolling three dice from a score of 200, there are 216 possible rolls. One roll is 1-1-1, for 1200 total points, or ER of 5.6. All 6's is 3.7, all 5's 3.2, 2.8, all 3's 2.3, and all 2'2 1.9.
We continue on: two 1's and a 5 will come up 3 ways, for ER of 3/216 x 450, or 6.2. Two 5's and a 1 is 5.6. Two 1's and an X (2, 3, 4, or 6) are 12 rolls, for ER of 22.2. Two 5's and an X is 16.7. 1-5-X is 24 rolls, for ER of 38.9. 1-X-X is 48 rolls, for ER of 66.7. 5-X-X puts you at 250 and you have to roll again; your result from below is 188.9, which will happen 48/216 of the time, for ER of 42.0.
Total ER for rolling 3 dice from 200 is 217.8.
When you have the choice of rolling 2 or 3 dice, you have to figure the result for rolling two dice at 250, which we haven't done yet. 1-1 has ER of 12.5, 5-5 og 9.7, 5-1 of 22.2, 1-X of 77.8, and 5-X of 66.7. This totals up to 188.9. Since this is less than the 250 you started out with, this means you would pick up the dice and not roll again if the rules allowed this.
Since 217.8 is higher than 188.9, we have shown that it is better to roll 3 dice from 200 than 2 dice from 250, i.e., pick up the five when you can.
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