Three balls numbered 1,2, and 3 are placed in a bag. A ball is drawn from the bag and the number is recorded. The ball is returned to the bag. After this has been done three times, what is the probability that the sum of the three recorded numbers is less than 8 ?
(A) $\frac{23}{27}$
(B) $\frac{5}{7}$
(C) $\frac{5}{9}$
(D) $\frac{8}{9}$
(E) $\frac{22}{27}$
Solution

Each time a ball is drawn from the bag, there are three possible outcomes. Since a ball is drawn three different times, there are $3^{3}=27$ possible outcomes for the sum, although not all 27 are unique.

To determine how many of these outcomes give a sum that is less than eight, we first determine how many give a sum of eight or nine.

The only one way the sum of the three recorded numbers could be nine, is if three 3's are drawn.

To yield a sum of 8 , the following three combinations are possible: $3,3,2$ or $3,2,3$ or 2,3 , 3.

Thus, of the 27 outcomes, $27-1-3=23$ give a sum less than 8 , so the probability of obtaining the required sum is $\frac{23}{27}$.

ANSWER: (A)

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