Difference between revisions of "2005 Alabama ARML TST Problems/Problem 1"
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==Problem== | ==Problem== | ||
| − | Two six-sided dice are constructed such that each face is equally likely to show up when rolled. The numbers on the faces of one of the dice are <math>1, 3, 4, 5, 6, and 8</math>. The numbers on the faces of the other die are <math>1, 2, 2, 3, 3, and 4</math>. Find the [[probability]] of rolling a sum of <math>9</math> with these two dice. | + | [[Two]] six-sided dice are constructed such that each face is equally likely to show up when rolled. The numbers on the faces of one of the dice are <math>1, 3, 4, 5, 6,\text{ and }8</math>. The numbers on the faces of the other die are <math>1, 2, 2, 3, 3,\text{ and }4</math>. Find the [[probability]] of rolling a sum of <math>9</math> with these two dice. |
==Solution== | ==Solution== | ||
Latest revision as of 00:30, 3 January 2023
Problem
Two six-sided dice are constructed such that each face is equally likely to show up when rolled. The numbers on the faces of one of the dice are 
. The numbers on the faces of the other die are 
. Find the probability of rolling a sum of 
with these two dice.
Solution
We use generating functions to represent the sum of the two dice rolls:
The coefficient of 
, that is, the number of ways of rolling a sum of 9, is thus 
, out of a total of 
possible two-roll combinations, for a probability of 
.
Alternatively, just note the possible pairs which work: 
and 
are all possible combinations that give us a sum of (where we count 
twice because there are two different 
s to roll). Thus the probability of one of these outcomes is 
.
See also
| 2005 Alabama ARML TST (Problems) | ||
| Preceded by: First question |
Followed by: Problem 2 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
