Diceand the legislations of Probability



EdwardD. Collins

For more questions and also problems about dice (and coin)probabilities, please seethis page.

You are watching: Probability of rolling at least one 6 in 4 rolls

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





Quitesome time ago, i was over at a friend"s home watching him and anotherfriend play a board game called Axis & Allies.At one allude this specific scenario come up - Kent to be planning top top rollingthree dice and reallywanted at the very least one 6 come appear. That made a comment that with threedice, his opportunities were 3/6 or 50%.

Kent"sreasoning was, with one die, the chances of roll a 6 were 1/6 i m sorry is correct. Healso thought if he were to roll 2 dice, his possibilities were dual thisor 2/6. This is incorrect andthis is whereby his faulty thinking begins.

Knowinga little bit around the regulations of probability, I quickly knew the fraction"2/6" for 2 dice and "3/6" for three dice was incorrect and also spent a short moment computing and then explaining the true percentages. Unfortunately, I execute notbelieve I prospered in explaining to Kent why my figures werecorrect. Probably I deserve to do therefore here. The knowledge obtained could definitely be really useful if you wish toplay cost-free craps games.

Obviously,with Kent"s reasonable above, if the chances of roll a 6 with two dice is2/6 and also the possibilities ofrolling a 6 with three dice is 3/6, then the chances of roll a 6with 6 dice would certainly be 6/6 !! 100%?? of course, this is obviouslyincorrect. I don"t treatment how many dice friend roll, the possibilities of rollinga 6 will never ever be 100%.

Whenyou roll simply one die, there space six different ways the die deserve to land,as displayed by the following graphic:


Whentwo dice room rolled, there are now 36 different and also unique methods thedice have the right to come up. This figure is arrived at by multiplying the numberof means the first die deserve to come up (six) by the variety of ways thesecond die can come up (six). 6 x 6 = 36.

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


Ifyou use the over graphic and count the number of times is 6 appearswhen two dice space rolled, friend will see the answer is eleven. Eleventimes the end of 36 or 30.5 %, slightly much less than the 33.3% (2/6) Kent thought. Whenyou roll 2 dice, you have actually a 30.5 % opportunity at least one 6 will appear.

Thisfigure can additionally be determined mathematically, without the usage of thegraphic. One means to carry out so is to take the variety of ways a solitary diewill NOT display a 6 as soon as rolled (five) and multiply this through the number ofways the second die will certainly NOT show a 6 when rolled. (Also five.) 5 x 5 =25. Subtract this from the total variety of ways 2 dice can appear(36) and we have actually our answer...eleven.

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

Takethe opportunities of a 6 NOT appearing on the an initial die...

5 / 6

andmultiply this by the opportunities of a 6 NOT appearing on the second die...

5 / 6 x 5 / 6 = 25 / 36

andmultiply this by the opportunities of a 6 NOT showing up on the 3rd die...

See more: How To Pop My Own Cherry ? Experts Weigh In How To Know If You Popped Your Cherry

25 / 36 x 5 / 6 = 125 / 216

So,there space 125 out of 216 opportunities of a 6 NOT showing up when three diceare rolled. Merely subtract 125 native 216 i beg your pardon will offer us the chancesa 6 WILL appear when 3 dice are rolled, i m sorry is 91. 91 the end of 216or 42.1 %, not rather the 50% Kent originally thought.

Hereis a table reflecting the fractions and percentages the a 6 showing up (orany other solitary digit for the matter) and notappearing v several different numbers of dice: