Difference between revisions of "2016 AMC 8 Problems/Problem 13"

Revision as of 00:17, 15 January 2024

Problem

Two different numbers are randomly selected from the set $\{ - 2, -1, 0, 3, 4, 5\}$ and multiplied together. What is the probability that the product is $0$ ?

$\textbf{(A) }\dfrac{1}{6}\qquad\textbf{(B) }\dfrac{1}{5}\qquad\textbf{(C) }\dfrac{1}{4}\qquad\textbf{(D) }\dfrac{1}{3}\qquad \textbf{(E) }\dfrac{1}{2}$

Solutions

Solution 1

The product can only be $0$ if one of the numbers is $0$ . Once we chose $0$ , there are $5$ ways we can chose the second number, or $6-1$ . There are $\dbinom{6}{2}$ ways we can chose $2$ numbers randomly, and that is $15$ . So, $\frac{5}{15}=\frac{1}{3}$ so the answer is $\boxed{\textbf{(D)} \, \frac{1}{3}}$ .

Solution 2 (Complementary Counting)

Because the only way the product of the two numbers is $0$ is if one of the numbers we choose is $0,$ we calculate the probability of NOT choosing a $0.$ We get $\frac{5}{6} \cdot \frac{4}{5} = \frac{2}{3}.$ Therefore our answer is $1 - \frac{2}{3} = \boxed{\textbf{(D)} \ \frac{1}{3}}.$

Solution 3 (Casework)

There are two different cases in which the product is zero; either the first number we select is zero, or the second one is. We consider these cases separately.

Case 1: 0 is the first number chosen

There is a $\frac{1}{6}$ chance of selecting zero as the first number. At this point, the product will be zero no matter the choice of the second number, so there is a $\frac{1}{6}$ of getting the desired product in this case.

Case 2: 0 is the second number chosen

There is a $\frac{5}{6}$ chance of choosing a number that is NOT zero as the first number. From there, there is a $\frac{1}{5}$ chance of picking zero from the remaining 5 numbers. Thus, there is a $\frac{5}{6} \cdot \frac{1}{5} = \frac{1}{6}$ chance of getting a product of 0 in this case.

Adding the probabilities from the two distinct cases up, we find that there is a $\frac{1}{6} + \frac{1}{6} = \boxed{\textbf{(D)} \ \frac{1}{3}}$ chance of getting a product of zero.

Video Solution (CREATIVE THINKING!!!)

https://youtu.be/cRsvq0BH4MI

~Education, the Study of Everything

Video Solution by OmegaLearn

https://youtu.be/6xNkyDgIhEE?t=357

~ pi_is_3.14

Video Solution

https://youtu.be/jDeS4A6N-nE

~savannahsolver

@@ Line 12: / Line 12: @@
 ===Solution 2 (Complementary Counting)===
 Because the only way the product of the two numbers is <math>0</math> is if one of the numbers we choose is <math>0,</math> we calculate the probability of NOT choosing a <math>0.</math> We get <math>\frac{5}{6} \cdot \frac{4}{5} = \frac{2}{3}.</math> Therefore our answer is <math>1 - \frac{2}{3} = \boxed{\textbf{(D)} \ \frac{1}{3}}.</math>
+===Solution 3 (Casework)===
+There are two different cases in which the product is zero; either the first number we select is zero, or the second one is. We consider these cases separately.
+Case 1: 0 is the first number chosen
+There is a <math>\frac{1}{6}</math> chance of selecting zero as the first number. At this point, the product will be zero no matter the choice of the second number, so there is a <math>\frac{1}{6}</math> of getting the desired product in this case.
+Case 2: 0 is the second number chosen
+There is a <math>\frac{5}{6}</math> chance of choosing a number that is NOT zero as the first number. From there, there is a <math>\frac{1}{5}</math> chance of picking zero from the remaining 5 numbers. Thus, there is a <math>\frac{5}{6} \cdot \frac{1}{5} = \frac{1}{6}</math> chance of getting a product of 0 in this case.
+Adding the probabilities from the two distinct cases up, we find that there is a <math>\frac{1}{6} + \frac{1}{6} = \boxed{\textbf{(D)} \ \frac{1}{3}}</math> chance of getting a product of zero.
 ==Video Solution (CREATIVE THINKING!!!)==

2016 AMC 8 (Problems • Answer Key • Resources)
Preceded by Problem 12	Followed by Problem 14
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