Suppose a pair of fair dice is tossed 3 times. Let $X$ be the sum of the outcomes at each toss. Find the probability of getting a sum of 7 (i.e., $X=7$) at least once out of the three tosses.

My attempt:

The sample space is $6^3=216.$ There are 15 combinations that gives a sum of $7$.


I am stuck here. If a pair of fair dice is tossed 3 times, should the sample space be 36 or 216? Am I on the right path?

  • 1
    $\begingroup$ You have to show how you worked out $\frac{15}{216}$. The method you used could have a problem , or it could be a calculation : either way, showing it would make life easier for somebody trying to answer your question. Did you enumerate the possibilities, or notice any independence etc.? $\endgroup$ – астон вілла олоф мэллбэрг Oct 11 '18 at 13:30
  • $\begingroup$ Hint: find the probability of the complement of that event, i.e. the probability that a sum $7$ does not arise at the $3$ tosses. $\endgroup$ – drhab Oct 11 '18 at 13:32
  • $\begingroup$ Your result $15/216$ accords with the probability that the sum of three dice rolled once (or one die rolled three times) will be $7$. $\endgroup$ – Barry Cipra Oct 11 '18 at 13:48

This is Bernouly process. The probability that you get 7 at one toss is $$p={6\over 36}={1\over 6}$$

So not getting 7 in 3 trials is $$P' = q^3 = (1-p)^3 = {125\over 216}$$

So the finaly answer is $$ P = 1-P' ={91\over 216}$$


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