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 | + | Since a ball is drawn three times and the sum is <math>6</math>, the only ways this can happen are the permutations of <math>1, 2, 3</math> and <math>2, 2, 2</math>. Therefore, with there being <math>3! + 1 = 7</math> equally-likely possibilities in total, the probability that the ball numbered <math>2</math> was drawn all three times is <math>\boxed{\textbf{(C)}\ \frac{1}{7}}</math>. |
| + | |||
| + | ==See Also== | ||
| + | {{AHSME box|year=1983|num-b=14|num-a=16}} | ||
| + | |||
| + | {{MAA Notice}} | ||
Latest revision as of 23:54, 19 February 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 ways this can happen are the permutations of 
and 
. Therefore, with there being 
equally-likely possibilities in total, the probability that the ball numbered was drawn all three times is 
.
See Also
| 1983 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 14 |
Followed by Problem 16 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
| All AHSME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
