Difference between revisions of "1983 AHSME Problems/Problem 15"
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==Problem== | ==Problem== | ||
| − | Three balls marked <math>1,2</math> | + | Three balls marked <math>1,2</math> and <math>3</math>, are placed in an urn. One ball is drawn, its number is recorded, and then the ball is returned to the urn. This process is repeated and then repeated once more, and each ball is equally likely to be drawn on each occasion. If the sum of the numbers recorded is <math>6</math>, what is the probability that the ball numbered <math>2</math> was drawn all three times? |
| − | then the ball is returned to the urn. This process is repeated and then repeated once more, | ||
| − | and each ball is equally likely to be drawn on each occasion. If the sum of the numbers recorded is <math>6</math>, | ||
| − | what is the probability that the ball numbered <math>2</math> was drawn all three times? | ||
<math> | <math> | ||
| − | \ | + | \textbf{(A)} \ \frac{1}{27} \qquad |
| − | \ | + | \textbf{(B)} \ \frac{1}{8} \qquad |
| − | \ | + | \textbf{(C)} \ \frac{1}{7} \qquad |
| − | \ | + | \textbf{(D)} \ \frac{1}{6} \qquad |
| − | \ | + | \textbf{(E)}\ \frac{1}{3} </math> |
==Solution== | ==Solution== | ||
Since a ball is drawn three times and the sum is <math>6</math>, the only possibilities of this happening are permutations of <math>1, 2, 3</math> and <math>2, 2, 2</math>. Therefore, there are 7 possibilities so the probability that the ball numbered <math>2</math> was drawn all three times would be <math>\boxed{\textbf{(C)}\ \frac{1}{7}}</math>. | Since a ball is drawn three times and the sum is <math>6</math>, the only possibilities of this happening are permutations of <math>1, 2, 3</math> and <math>2, 2, 2</math>. Therefore, there are 7 possibilities so the probability that the ball numbered <math>2</math> was drawn all three times would be <math>\boxed{\textbf{(C)}\ \frac{1}{7}}</math>. | ||
Revision as of 19:55, 26 January 2019
Problem
Three balls marked 
and 
, are placed in an urn. One ball is drawn, its number is recorded, and then the ball is returned to the urn. This process is repeated and then repeated once more, and each ball is equally likely to be drawn on each occasion. If the sum of the numbers recorded is 
, what is the probability that the ball numbered 
was drawn all three times?
Solution
Since a ball is drawn three times and the sum is , the only possibilities of this happening are permutations of 
and 
. Therefore, there are 7 possibilities so the probability that the ball numbered was drawn all three times would be 
.
