I'm having a talk with my mate regarding a table top game (warhammer in this case) and how much he will be relying on a small % chance of something happening. Statistics are not my friend so I thought some smart bugger can settle this.
What is the probability of the following happening at least once, assuming all dice are 6 sided.
Start by rolling 4 dice, keep any number of them that show 2 or more, discard any 1s.
Roll the left over dice, keep any number of them that show a 6, re roll any 1s that are showing applying this rule to it once more but ignoring the reroll if a further 1 is rolled part.
Take any left over dice and roll them, any number between 1 and 4 show up keep those.
Finish.
So yeah what's the chance of all of that happening at least once? My mind melts after the first stage.
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So P(one dice completing all tasks):
P(not a 1)*P(6 with possible reroll)*P(between 1 and 4)
The first and the last probabilities are trivial.
The second is P(6 w/ reroll) = P(6) or P(1 and then 6)
These are disjoint events so: P(6 w/ reroll) = 1/6 + 1/6*1/6 = 7/36
So
P(completing task) = 5/6*7/36*4/6 = .1080 , or 10.80%
The probability of at least one dice out of four completing task is the same as 1 minus P(no dice making it), so
1 - P(no success in four dice) = 1 - (1-.1080)^4 = .3667 or 36.70%
Nintendo ID: Pastalonius
Smite\LoL:Gremlidin \ WoW & Overwatch & Hots: Gremlidin#1734
3ds: 3282-2248-0453
P(0 success) = .6331
P(1 success) = .3066
P(2 success) = .0557
P(3 success) = .0045
P(4 success) = .0001
Nintendo ID: Pastalonius
Smite\LoL:Gremlidin \ WoW & Overwatch & Hots: Gremlidin#1734
3ds: 3282-2248-0453
Zombie Hero, thanks for the working more then anything. Means I can reproduce it in similar but different situations!
Like:
Step 1, remove any dice that are 1s from the pool
Step 2a, roll remaining dice, Put 6s aside
Step 2b, reroll any dice from step 2a that were 1s, if 6s Put aside?
Step 3, reroll any remaining dice, do what with dice 1-4?
Like are we looking for a sum after step 3? Including the 6s set aside in step 2? Or just the number of dice that are between 1-4 in step 3? Or that number + the number of 6s from step 2?
Step 1, remove any dice that are 1s from the pool
Step 2a, roll remaining dice, remove any dice that are not 6s from the pool
Step 2b, reroll any dice from step 2a that were 1s, remove any dice that are not 6s from the pool
Step 3, reroll any remaining dice, values of 1-4 are successes
Does that make it clearer? Im after the prob of at least 1 success.
Step 1: 5/6 = 83,3%
Step 2: 1/6 + 1/36 = 19.4%
Step 3: 4/6 = 66.7%
Total: 0.833 * 0.194 * 0.667 = 10.7%
It follows that the probability of failure with just 1 die would be 89.3%. Now, let's look at the 4 dice. The probability of at least one of them passing the trials is (1 - the probability that none of them passes):
P(total) = 1 - 0.893^4 = 36.4%
So that's the probability that you get to keep at least one of the die rolls in the end.
edit: beat'd.
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