I was calculating the odds of a set of dice rolls in a game of Catan recently. In 35 rolls, we rolled 0 3's, and 0 5's.
To calculate the probability of this, I calculated the probability of rolling 0 3's in 35 rolls, and multiplied by the probability of rolling 0 5's in 35 rolls:
P(3)^0 * P(not 3)^35 * P(5)^0 * P(not 5)^35 = (2/36)^0 * (34/36)^35 * (4/36)^0 * (32/36)^35 = 2.19 E -3
But then I thought: this calculated probability should equal the probability of rolling a "not 3 or 5", 35 times:
P(not 3 or 5)^35 = (30/36)^35 = 1.69 E -3
These two different approaches do not yield the same answer. I've struggled to reason through which approach is correct. I've thought that the answers don't agree because "zero" is the roll; that it makes sense to say "roll two (3 or 5)'s in ten rolls" (in which case you'd use the second formula) but, if talking about "zero", must you say zero 3's AND 5's, and use the first formula?
A commenter on another site says that "rolling 0 3's makes a 5 more likely, thus rolling a 3 and rolling a 5 are not independent". I am not sure that I buy that answer; rolling a 3 has no bearing on rolling a 5, thus how could they not be independent? I can roll 100 3's in a row, or none, and that has no bearing on whether my next roll will be a 5 or not, right? If someone could flesh out the concept a little more, I'd be very grateful- after all, Catan-bragging-rights are on the line.