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Some probability problems....Cosmic Wimpout Style...
blaisepascal
Imagine I have 5 dice. What's the probability that on rolling the dice at least one die will show a 1 or a 5?

This is a relatively easy problem, well, easy enough once you invert it: the probability at least one die will show a 1 or a 5 is 1, less the probability that no die shows a 1 or a 5. Since there are 4 ways for a die to not show a 1 or a 5, there's a 2/3 chance, per die, or (2/3)5 that none will. That's 32/243, or 13.2%. So there's a 86.8% chance of getting a 1 or a 5 on at least one die.

We can do that for smaller numbers of dice as well: For 1 die, 33.3%; for 2 dice, 55.6%; for 3 dice, 70.4%; for 4 dice, 80.2%; and (to repeat) for 5 dice, 86.8%.

OK, for the next complication.... One of my 5 dice is a different color than the others. Let's change the definition of a successful roll from 1 or 5 on any die to 1 or 5 on any die or a 3 on the special die (a 1 or a 2 works on that die as well). What's the probability of succeeding this time?

We can see that there is a (2/3)4 chance of none of the normal dice getting a 1 or a 5, and there's a 1/2 chance that the special die won't show a 1 or a 5. So the overall chance of failing is (2/3)4(1/2), or 16/162 (9.9%), so there's a 90.1% chance of success. Varying the number of dice: 1 special: 50%, 1 special, 1 normal: 66.6%, 2 normal: 77.8%, 3 normal: 85.2%, 4 normal 90.1%.

What's this have to do with Cosmic Wimpout? For that matter, what is Cosmic Wimpout?

Cosmic Wimpout is a dice game, played with 5 specially marked dice. On four of them (white in color), there are pips for 2, 3, 4, and 6 on 4 of the sides. The other two sides are marked 5 and 10. The fifth, black, die, is marked nearly the same. Instead of 3 pips on one side, it has a sun icon. The object of the game is to roll the dice to try to score points -- the 5's and 10's. If you roll a 5 or a 10 (or the sun icon, which is considered "wild") on any die you roll, you score that many points, and can (optionally) reroll the non-scoring dice if you choose. If you choose not to reroll nonscoring dice, your turn ends, and you get credited with all the points scored that turn. If you reroll, and fail to score, you "wimp out", lose all the points rolled this turn, and your turn ends.

There are more rules I'll talk about in another posting, but that's the basic idea. The other rules add more ways of scoring, but also add more rolling requirements (such as the rule "You may not want to, but you must", which states that if you are scoring on all 5 dice, you must reroll all 5 dice, thus risking losing all of them).

The probability problems above correspond to the ways to score in Cosmic Wimpout. The chances of scoring when rolling all five dice is 90.1%. Usually, you can choose to quit then; but should you? It depends... If you scored on 1 white die, you have a 1/7 chance of wimping out (roughly). 1 black die, it rises to 1/5. Two white dice, 2/9. a white and black die, 3/10. three white: 1/3, a black and 2 white: 4/9, 4 whites: 1/2, and a black and 3 whites 2/3. Pick the cutoff you want, and go with it.

The actual rules make it more complicated, of course. I'll examine those cases at a later time.

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