Sure! Let's break down the problem step-by-step:
1. Understanding the Problem:
- Total number of candies = 36
- Probability of picking a lemon candy [tex]\(\frac{1}{3}\)[/tex]
2. Determine the Initial Number of Lemon and Orange Candies:
- To find the number of lemon candies, we use the given probability:
[tex]\[
\frac{\text{Number of lemon candies}}{\text{Total number of candies}} = \frac{1}{3}
\][/tex]
Given the total candies are 36, let's denote the number of lemon candies as [tex]\( L \)[/tex]:
[tex]\[
\frac{L}{36} = \frac{1}{3}
\][/tex]
[tex]\[
L = 36 \times \frac{1}{3} = 12 \text{ lemon candies}
\][/tex]
- Then, the number of orange candies is:
[tex]\[
36 - 12 = 24 \text{ orange candies}
\][/tex]
3. Set up the Equation to Find How Many Lemon Candies to Add:
- Let's denote the number of lemon candies to add as [tex]\( x \)[/tex].
- After adding [tex]\( x \)[/tex] lemon candies, the new number of lemon candies will be [tex]\( 12 + x \)[/tex].
- The total number of candies will now be [tex]\( 36 + x \)[/tex].
- We want the probabilities of picking a lemon candy and an orange candy to be equal, which means:
[tex]\[
\frac{12 + x}{36 + x} = \frac{24}{36 + x}
\][/tex]
- At this point, we have two equal fractions with the same denominator:
[tex]\[
12 + x = 24
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
- To find [tex]\( x \)[/tex], simply solve the equation:
[tex]\[
12 + x = 24
\][/tex]
[tex]\[
x = 24 - 12
\][/tex]
[tex]\[
x = 12
\][/tex]
Therefore, you need to add 12 lemon candies to make the probability of picking a lemon candy equal to the probability of picking an orange candy.
The correct answer is E. 12.
