# What is the probability that the sum will be less than $6$ when three fair six-sided dice are thrown? [closed]

Three fair six-sided dice are thrown, and the score is the sum of the three results. What is the probability that the score is less than $6$?

## closed as off-topic by Namaste, Matthew Conroy, JMoravitz, TravisJ, N. F. TaussigMay 1 '17 at 20:01

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• What are the different ways to add up to $3$, $4$, or $5$, if you count $1+1+2$ as different from $1+2+1$ (say)? How many different ways are there to roll three dice altogether? – Brian Tung May 1 '17 at 19:23
• Are the throws hidden or right before the eyes of the presumed player? – mathreadler May 1 '17 at 19:35
• If you know about random variables you can get a close approximation by assuming the sum on three rolls is $S = X_1 + X_2 + X_3,$ where $X_i$ is the result on the $i$th die, $X_i$ independent. Then $E(X_i) = 3.5,$ $V(X_i)=8.75, E(S) = 10.5,$ and $SD(S) = 2.96.$ Assuming $S\stackrel{aprx}{\sim}\mathsf{Norm}(10.5,2.96),$ you can get $P(S < 5.5)\approx 0.046,$ which is very nearly the exact answer $10/216$ obtained by counting outcomes. (The normal approximation is 'pretty good' in the tails of the distribution.) – BruceET May 1 '17 at 20:55

Let $P(X<6)$, where $X =$ the sum of $x_1, x_2, x_3$

In this case, we can simply list out all the outcome such $P(X<6)$

1. $(1,1,1), (1,1,2), \dots, (1,1,3), \dots, (3,1,1).$

Note that each event has the same probability:

1. Find the probability of one event: $P(x_1 \cap x_2 \cap x_3)$

Connecting both 1. and 2. you should be able to go on to the next and final step to solving the problem.

• You need more dots (and commas) on line 1. to indicate more clearly it is only a partial list. I have made a minor edit. (+1) – BruceET May 1 '17 at 19:56
• Thanks Bruce. I'm new to mathexchange, as well as writing mathematical text online. – man moon May 1 '17 at 20:01