Simulation vs Theory: Rolling a Sum of 7

0:00Roll two dice and add them: 3 and 4 make 7. How often should the sum be 7?

0:08A common mistake is to say the 11 sums, from 2 to 12, each have a chance of 1 in 11.

0:15An outcome is one possible result. This table shows every outcome, with the sum of the two dice in each cell.

0:246 times 6 gives 36 outcomes, all equally likely with fair dice.

0:30Now count the cells with a sum of 7: 1, 2, 3, 4, 5, 6.

0:38A sum of 2 appears only once, so the sums are not equally likely.

0:44Theoretical probability comes from reasoning. For equally likely outcomes, it is the outcomes we want over all outcomes.

0:53That is 6 over 36, or one sixth, about 0.167.

1:00Experimental probability comes from trials. A trial is one roll of both dice, and a simulation runs many trials on a computer.

1:10A simulation of 200 rolls gave 31 sevens, so the experimental probability is 31 over 200, or 0.155.

1:22That is close to 0.167, but not equal.

1:27They differ because each roll is random. In 200 rolls, one seven moves the result by 0.005.

1:36A second simulation of 2000 rolls gave 336 sevens. 336 over 2000 is 0.168, much closer to 0.167.

1:50In 2000 rolls, one seven moves the result by only 0.0005.

1:57With more trials, random ups and downs even out, so the result usually gets closer to the theory.

2:05Recap: theory gives 6 over 36, about 0.167, and more trials usually bring a simulation closer.