Diceand the regulations of Probability



EdwardD. Collins

For an ext questions and also problems concerning dice (and coin)probabilities, please seethis page.

You are watching: Odds of rolling a 6 with 2 dice

Let"simagine you are playing a video game which uses dice. Friend are around to rollthree the them. You must roll at the very least one 6. A 6 showing up onany one (or more) the the 3 dice will win the video game for you! What areyour chances?





Quitesome time ago, ns was end at a friend"s residence watching him and also anotherfriend play a board game called Axis & Allies.At one point this exact scenario came up - Kent to be planning on rollingthree dice and also reallywanted at least one 6 come appear. That made a comment that with threedice, his opportunities were 3/6 or 50%.

Kent"sreasoning was, v one die, the chances of roll a 6 were 1/6 i m sorry is correct. Healso thought if he to be to roll two dice, his chances were double thisor 2/6. This is untrue andthis is whereby his faulty reasoning begins.

Knowinga tiny bit about the laws of probability, I conveniently knew the fraction"2/6" for two dice and "3/6" for three dice was incorrect and also spent a brief moment computing and also then explaining the true percentages. Unfortunately, I carry out notbelieve I was successful in explaining to Kent why my numbers werecorrect. Probably I have the right to do so here. The knowledge got could absolutely be very useful if you great toplay cost-free craps games.

Obviously,with Kent"s logic above, if the possibilities of roll a 6 through two dice is2/6 and the possibilities ofrolling a 6 with 3 dice is 3/6, then the chances of roll a 6with 6 dice would certainly be 6/6 !! 100%?? of course, this is obviouslyincorrect. Ns don"t care how numerous dice girlfriend roll, the opportunities of rollinga 6 will never be 100%.

Whenyou roll just one die, there are six different ways the die deserve to land,as displayed by the adhering to graphic:


Whentwo dice space rolled, there are now 36 different and also unique means thedice deserve to come up. This figure is came down on by multiplying the numberof ways the very first die can come increase (six) by the variety of ways thesecond die can come increase (six). 6 x 6 = 36.

Thisgraphic mirrors this really nicely. I"ve offered two different colored die tohelp present a roll of 2-1 is different from a role of 1-2.


Ifyou usage the above graphic and count the variety of times is 6 appearswhen two dice are rolled, you will check out the prize is eleven. Eleventimes the end of 36 or 30.5 %, slightly less than the 33.3% (2/6) Kent thought. Whenyou roll two dice, you have a 30.5 % possibility at the very least one 6 will certainly appear.

Thisfigure can also be figured out mathematically, without the usage of thegraphic. One method to do so is to take it the variety of ways a single diewill NOT display a 6 as soon as rolled (five) and multiply this by the number ofways the 2nd die will certainly NOT show a 6 when rolled. (Also five.) 5 x 5 =25. Subtract this from the total variety of ways two dice can appear(36) and also we have our answer...eleven.

So,let"s use this same an approach to answer ours question and determine thechances of at least one 6 showing up when three diceare rolled.

Takethe chances of a 6 NOT appearing on the first die...

5 / 6

andmultiply this through the chances of a 6 NOT showing up on the 2nd die...

5 / 6 x 5 / 6 = 25 / 36

andmultiply this by the possibilities of a 6 NOT appearing on the 3rd die...

See more: What Book In The Bible Contains The Longest Verse S Of The Bible

25 / 36 x 5 / 6 = 125 / 216

So,there room 125 out of 216 opportunities of a 6 NOT showing up when three diceare rolled. Simply subtract 125 indigenous 216 i m sorry will offer us the chancesa 6 WILL show up when three dice are rolled, i m sorry is 91. 91 the end of 216or 42.1 %, not fairly the 50% Kent originally thought.

Hereis a table mirroring the fractions and also percentages that a 6 showing up (orany other solitary digit for that matter) and also notappearing v several various numbers the dice: