What is the probability of rolling a 6 symbol combination using 8 identical 6-sided dice (each dice having one duplicate symbol ie. d={A,A,B,C,D,E}), where the player is allowed discard one dice in order to reroll up to seven remaining?
Notes:
– The game has eight identical six-sided dice. Each dice has five
symbols, the sixth symbol is a duplicate ie. d={A,A,B,C,D,E}.
– The player is allowed a one-time reroll up to seven of her dice (but it costs a one dice discard)
– The goal is to match a six-symbol combination (I have linked an example below).
To be honest, I don't know where to begin. I have worked out that there are 210 possible combinations of 6-symbols.
I.e. n=5 symbols and r=6
$$\frac{(r+n-1)!}{r!(n-1)!} = \frac{10!}{6!4!} = 210$$
Example of a symbol combination
I really can't figure out how to calculate the probability though.
Best Answer
Although there is surely a closed-form solution as @BGM alluded to, you can also simulate the situation. Following is code to simulate a sampling distribution.
In my run of the simulation, the mean probability of getting a 6 character match was $1.959$% with standard deviation $0.397$%. Because the standard error is the standard deviation of the sampling distribution, in 95% of repetitions of 100 sets of 8 rolls with one being replaced, between 1.18% and 2.74% will have a 6-character match. Of course this assumes a random replacement roll, but most people would try to behave strategically. However, I doubt it would change the results by very much. Additionally, this assumes the rolls are normally distributed. Visual analysis says the rolls are normally distributed.