Difference between revisions of "2007 AMC 12A Problems/Problem 12"
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+ | {{duplicate|[[2007 AMC 12A Problems|2007 AMC 12A #12]] and [[2007 AMC 10A Problems/Problem 16|2007 AMC 10A #16]]}} | ||
==Problem== | ==Problem== | ||
Integers <math>a, b, c,</math> and <math>d</math>, not necessarily distinct, are chosen independently and at random from 0 to 2007, inclusive. What is the [[probability]] that <math>ad-bc</math> is [[even]]? | Integers <math>a, b, c,</math> and <math>d</math>, not necessarily distinct, are chosen independently and at random from 0 to 2007, inclusive. What is the [[probability]] that <math>ad-bc</math> is [[even]]? | ||
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==Solution== | ==Solution== | ||
− | The only times when <math>ad-bc</math> is even is when <math>ad</math> and <math>bc</math> are of the same [[parity]]. The chance of <math>ad</math> being odd is <math>\frac 12 \cdot \frac 12 = \frac 14</math>, so it has a <math>\frac 34</math> probability of being even. Therefore, the probability that <math>ad-bc</math> will be even is <math> | + | The only times when <math>ad-bc</math> is even is when <math>ad</math> and <math>bc</math> are of the same [[parity]]. The chance of <math>ad</math> being odd is <math>\frac 12 \cdot \frac 12 = \frac 14</math>, so it has a <math>\frac 34</math> probability of being even. Therefore, the probability that <math>ad-bc</math> will be even is <math>\left(\frac 14\right)^2+\left(\frac 34\right)^2=\frac 58\ \mathrm{(E)}</math>. |
==See also== | ==See also== | ||
{{AMC12 box|year=2007|ab=A|num-b=11|num-a=13}} | {{AMC12 box|year=2007|ab=A|num-b=11|num-a=13}} | ||
+ | {{AMC10 box|year=2007|ab=A|num-b=15|num-a=17}} | ||
[[Category:Introductory Combinatorics Problems]] | [[Category:Introductory Combinatorics Problems]] |
Revision as of 18:50, 5 January 2008
- The following problem is from both the 2007 AMC 12A #12 and 2007 AMC 10A #16, so both problems redirect to this page.
Problem
Integers and , not necessarily distinct, are chosen independently and at random from 0 to 2007, inclusive. What is the probability that is even?
Solution
The only times when is even is when and are of the same parity. The chance of being odd is , so it has a probability of being even. Therefore, the probability that will be even is .
See also
2007 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 11 |
Followed by Problem 13 |
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 | |
All AMC 12 Problems and Solutions |
2007 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 15 |
Followed by Problem 17 | |
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 | ||
All AMC 10 Problems and Solutions |