Dice probability help

[Morrus]

My brain is refusing to work.

What is the probability of rolling 2 or more sixes on 2d6, 3d6, 4d6, 5d6, 6d6.... 20d6?

What is the probability of rolling 3 or more sixes on same?

I can manage the first bit - on 2 or more 6s on 2d6 is 1/36 or 2.8%. I get stuck once I add additional dice though. I'm sure it's very simple!

 

[Nikosandros]

In cases like this it's usually easier to calculate the probability of the complement and the subtract it from 1.

The probability of rolling no sixes on N dice is P1=(5/6)^N. The probability of rolling a single six is P2=N(1/6)(5/6)^(N-1). The probability you're looking for is 1-(P1+P2).

 

[Nagol]

The probability of rolling exactly X alternative in Y attempts is probability of alternative X occurring to the power of the X multiplied by (1- probability of alternative X occurring) to the power of Y - X multiplied by Y! / ((Y-X)! * X!) (if using a spreadsheet, this last term is the Combin function).

So the probability of rolling 2 sixes on 2d6 is (1/6)**2 * (5/6)** 0 * 1 = 1/ 36
The probability of rolling 2 sixes on 3d6 is (1/6)**2 * (5/6)**1 * (3! / (2! * 1!) = 15 / 216.

To get the probability of a range of results (at least 2 6s...) either calculate the probability of each individual case and add them together or calculate the probabilities of all the cases that don't apply and subtract from 1.

 

[Morrus]

In cases like this it's usually easier to calculate the probability of the complement and the subtract it from 1.

The probability of rolling no sixes on N dice is P1=(5/6)^N. The probability of rolling a single six is P2=N(1/6)(5/6)^(N-1). The probability you're looking for is 1-(P1+P2).

Hmm. It sounds like I could work that up in a spreadsheet!

 

[Nagol]

It is quick to set up an individual row

if D7 is the chance of success on a single die, E7 is the target number of successes and C7 is the size of the dice pool then

=POWER(D7;E7)*POWER(1-D7;C7-E7)*COMBIN(C7;E7) will give you the probability. Note the formula is structured for OpenOffice. For Excel replace the semicolons with commas and it should be fine.