Difference between revisions of "AoPS Wiki talk:Problem of the Day/June 27, 2011"
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==Solutions== | ==Solutions== | ||
− | {{ | + | The product will be a multiple of <math>5</math> only when the top is a multiple of <math>5</math>. |
+ | |||
+ | We will divide this into three cases. | ||
+ | |||
+ | '''CASE 1: THE 5 APPEARS ON THE LEFT DIE BUT NOT THE RIGHT''' | ||
+ | |||
+ | There is a <math>\dfrac{1}{6}</math> chance of the left die getting the <math>5</math>, and a <math>\dfrac{5}{6}</math> chance of the right die not getting a <math>5</math>, so the probability that both happen is <math>\dfrac{1}{6}\times\dfrac{5}{6}=\dfrac{5}{36}</math>. | ||
+ | |||
+ | '''CASE 2: THE 5 APPEARS ON THE RIGHT DIE BUT NOT THE LEFT''' | ||
+ | |||
+ | We can follow the same logic here. There is a <math>\dfrac{1}{6}</math> chance of the right die getting the <math>5</math>, and a <math>\dfrac{5}{6}</math> chance of the left die not rolling a <math>5</math>, so the probability that both happen is <math>\dfrac{1}{6}\times\dfrac{5}{6}=\dfrac{5}{36}</math>. | ||
+ | |||
+ | '''CASE 2: THE 5 APPEARS ON BOTH DIE''' | ||
+ | |||
+ | The probability this happens is <math>\dfrac{1}{6}\times\dfrac{1}{6}=\dfrac{1}{36}</math>. (Do you see why?) | ||
+ | |||
+ | Adding these probabilities together, we find that the answer is <math>\dfrac{5}{6}+\dfrac{5}{6}+\dfrac{1}{6}=\boxed{\dfrac{11}{36}}</math>. |
Revision as of 20:32, 26 June 2011
Problem
AoPSWiki:Problem of the Day/June 27, 2011
Solutions
The product will be a multiple of only when the top is a multiple of .
We will divide this into three cases.
CASE 1: THE 5 APPEARS ON THE LEFT DIE BUT NOT THE RIGHT
There is a chance of the left die getting the , and a chance of the right die not getting a , so the probability that both happen is .
CASE 2: THE 5 APPEARS ON THE RIGHT DIE BUT NOT THE LEFT
We can follow the same logic here. There is a chance of the right die getting the , and a chance of the left die not rolling a , so the probability that both happen is .
CASE 2: THE 5 APPEARS ON BOTH DIE
The probability this happens is . (Do you see why?)
Adding these probabilities together, we find that the answer is .