Solve the following problem and choose the best answer:

A bag contains 15 red candies, 15 yellow candies, and 15 blue candies. What is the probability of randomly choosing two candies of the same color, without replacement?

A. 13
B. 19
C. 722
D. 115



Answer :

Let's solve the problem step by step:

### Step 1: Understand the Problem

We have a bag containing three different colors of candies:
- 15 red candies
- 15 yellow candies
- 15 blue candies

We need to find the probability of drawing two candies of the same color in a row without replacement.

### Step 2: Determine the Total Number of Candies

Total number of candies = 15 (red) + 15 (yellow) + 15 (blue) = 45 candies

### Step 3: Calculate the Probability of Drawing Two Candies of the Same Color

We will calculate the probability for each color individually and then sum these probabilities to get the total probability.

#### Probability of drawing two red candies:

- Probability of drawing the first red candy: [tex]\( \frac{15}{45} \)[/tex]
- Probability of drawing the second red candy after the first one has been taken out: [tex]\( \frac{14}{44} \)[/tex]

So, the probability of drawing two red candies is:
[tex]\[ P(\text{Red}) = \frac{15}{45} \times \frac{14}{44} \][/tex]

#### Probability of drawing two yellow candies:

- Probability of drawing the first yellow candy: [tex]\( \frac{15}{45} \)[/tex]
- Probability of drawing the second yellow candy after the first one has been taken out: [tex]\( \frac{14}{44} \)[/tex]

So, the probability of drawing two yellow candies is:
[tex]\[ P(\text{Yellow}) = \frac{15}{45} \times \frac{14}{44} \][/tex]

#### Probability of drawing two blue candies:

- Probability of drawing the first blue candy: [tex]\( \frac{15}{45} \)[/tex]
- Probability of drawing the second blue candy after the first one has been taken out: [tex]\( \frac{14}{44} \)[/tex]

So, the probability of drawing two blue candies is:
[tex]\[ P(\text{Blue}) = \frac{15}{45} \times \frac{14}{44} \][/tex]

### Step 4: Sum the Individual Probabilities

We add the probabilities of each color to get the total probability of drawing two candies of the same color:

[tex]\[ P(\text{same color}) = P(\text{Red}) + P(\text{Yellow}) + P(\text{Blue}) \][/tex]
[tex]\[ P(\text{same color}) = \left( \frac{15}{45} \times \frac{14}{44} \right) + \left( \frac{15}{45} \times \frac{14}{44} \right) + \left( \frac{15}{45} \times \frac{14}{44} \right) \][/tex]

### Step 5: Calculate the Numerical Value

By simplifying the values, we get:

[tex]\[ P(\text{same color}) = 3 \times \left( \frac{15}{45} \times \frac{14}{44} \right) \][/tex]
[tex]\[ P(\text{same color}) = 3 \times \left( \frac{15}{45} \times \frac{14}{44} \right) \][/tex]
[tex]\[ P(\text{same color}) = 3 \times \left( \frac{1}{3} \times \frac{7}{22} \right) \][/tex]
[tex]\[ P(\text{same color}) = 3 \times \frac{7}{66} \][/tex]
[tex]\[ P(\text{same color}) = \frac{21}{66} \][/tex]
[tex]\[ P(\text{same color}) = \frac{7}{22} \][/tex]

### Step 6: Convert the Probability to a Decimal

[tex]\[ P(\text{same color}) \approx 0.3181818181818181 \][/tex]

### Step 7: Simplify the Fraction

The simplified fraction for the probability [tex]\( \frac{7}{22} \)[/tex] in decimal form roughly translates to [tex]\( 0.3181818181818181 \)[/tex].

### Step 8: Choose the Correct Answer

Given the options provided, none directly match the simplified fraction or decimal form we found. As [tex]\( \frac{1432963517799703}{4503599627370496} \)[/tex], the answer which approximates to [tex]\( 0.3181818181818181 \)[/tex], most closely relates to the complexity and nature of the options provided compared to simpler fractional forms.

Thus, the best answer amongst the options is:

[tex]\[ \boxed{ \frac{7}{22} } \][/tex] (107/336 based on simplified form calculated)

Note that the overarching exact fraction based directly may seem slightly more complex [tex]\( \boxed{\frac{1432963517799703}{4503599627370496}} \)[/tex].