Difference between revisions of "2019 AMC 8 Problems/Problem 18"

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==Solution 1==
 
==Solution 1==
We have a <math>2</math> die with <math>2</math> evens and <math>4</math> odds on both dies. For the sum to be even, the 2 rolls be <math>2</math> odds or <math>2</math> evens.  
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We have <math>2</math> dice with <math>2</math> evens and <math>4</math> odds on each die. For the sum to be even, the 2 rolls must be <math>2</math> odds or <math>2</math> evens.  
  
Ways to roll <math>2</math> odds (Case <math>1</math>): The total number of ways to obtain <math>2</math> odds is <math>4*4=16</math>, as there are <math>4</math> possible odds on the first roll and <math>4</math> possible odds on the second roll.
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Ways to roll <math>2</math> odds (Case <math>1</math>): The total number of ways to obtain <math>2</math> odds on 2 rolls is <math>4*4=16</math>, as there are <math>4</math> possible odds on the first roll and <math>4</math> possible odds on the second roll.
  
Ways to roll <math>2</math> evens (Case <math>2</math>): Similarly, we have <math>2*2=4</math> ways to roll <math>{36}=\frac{20}{36}=\frac{5}{9}</math>, or <math>\framebox{C}</math>.
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Ways to roll <math>2</math> evens (Case <math>2</math>): Similarly, we have <math>2*2=4</math> ways to obtain 2 evens. Probability is <math>\frac{20}{36}=\frac{5}{9}</math>, or <math>\framebox{C}</math>.
  
 
==Solution 2 (Complementary Counting)==
 
==Solution 2 (Complementary Counting)==
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==Video Solution==
 
==Video Solution==
https://youtu.be/8fF55uF64mE - Happytwin
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https://youtu.be/8fF55uF64mE  
  
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- Happytwin
  
Associated video - https://www.youtube.com/watch?v=EoBZy_WYWEw
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https://www.youtube.com/watch?v=_IK58KFUYpk  ~David
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https://www.youtube.com/watch?v=EoBZy_WYWEw
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 +
Associated video
  
 
https://www.youtube.com/watch?v=H52AqAl4nt4&t=2s
 
https://www.youtube.com/watch?v=H52AqAl4nt4&t=2s
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== Video Solution ==
 
== Video Solution ==
  
Solution detailing how to solve the problem: https://www.youtube.com/watch?v=94D1UnH7seo&list=PLbhMrFqoXXwmwbk2CWeYOYPRbGtmdPUhL&index=19
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Solution detailing how to solve the problem:  
 +
 
 +
https://www.youtube.com/watch?v=94D1UnH7seo&list=PLbhMrFqoXXwmwbk2CWeYOYPRbGtmdPUhL&index=19
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==Video Solution==
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https://youtu.be/8gl4rCZMUFI
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~savannahsolver
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==Video Solution (CREATIVE THINKING!!!)==
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https://youtu.be/zrmHM_l1UkI
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~Education, the Study of Everything
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==See Also==
 
==See Also==
 
{{AMC8 box|year=2019|num-b=17|num-a=19}}
 
{{AMC8 box|year=2019|num-b=17|num-a=19}}
  
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 19:26, 15 April 2023

Problem 18

The faces of each of two fair dice are numbered $1$, $2$, $3$, $5$, $7$, and $8$. When the two dice are tossed, what is the probability that their sum will be an even number?

$\textbf{(A) }\frac{4}{9}\qquad\textbf{(B) }\frac{1}{2}\qquad\textbf{(C) }\frac{5}{9}\qquad\textbf{(D) }\frac{3}{5}\qquad\textbf{(E) }\frac{2}{3}$

Solution 1

We have $2$ dice with $2$ evens and $4$ odds on each die. For the sum to be even, the 2 rolls must be $2$ odds or $2$ evens.

Ways to roll $2$ odds (Case $1$): The total number of ways to obtain $2$ odds on 2 rolls is $4*4=16$, as there are $4$ possible odds on the first roll and $4$ possible odds on the second roll.

Ways to roll $2$ evens (Case $2$): Similarly, we have $2*2=4$ ways to obtain 2 evens. Probability is $\frac{20}{36}=\frac{5}{9}$, or $\framebox{C}$.

Solution 2 (Complementary Counting)

We count the ways to get an odd. If the sum is odd, then we must have an even and an odd. The probability of an even is $\frac{1}{3}$, and the probability of an odd is $\frac{2}{3}$. We have to multiply by $2!$ because the even and odd can be in any order. This gets us $\frac{4}{9}$, so the answer is $1 - \frac{4}{9} = \frac{5}{9} = \boxed{(\textbf{C})}$.

Solution 3

To get an even, you must get either 2 odds or 2 evens. The probability of getting 2 odds is $\frac{4}{6} * \frac{4}{6}$. The probability of getting 2 evens is $\frac{2}{6} * \frac{2}{6}$. If you add them together, you get $\frac{16}{36} + \frac{4}{36}$ = $\boxed{(\textbf{C})  \frac{5}{9}}$.

Video Solution

https://youtu.be/8fF55uF64mE

- Happytwin

https://www.youtube.com/watch?v=_IK58KFUYpk ~David

https://www.youtube.com/watch?v=EoBZy_WYWEw

Associated video

https://www.youtube.com/watch?v=H52AqAl4nt4&t=2s

Video Solution

Solution detailing how to solve the problem:

https://www.youtube.com/watch?v=94D1UnH7seo&list=PLbhMrFqoXXwmwbk2CWeYOYPRbGtmdPUhL&index=19

Video Solution

https://youtu.be/8gl4rCZMUFI

~savannahsolver

Video Solution (CREATIVE THINKING!!!)

https://youtu.be/zrmHM_l1UkI

~Education, the Study of Everything

See Also

2019 AMC 8 (ProblemsAnswer KeyResources)
Preceded by
Problem 17
Followed by
Problem 19
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 AJHSME/AMC 8 Problems and Solutions

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