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Sagot :
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].
### 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].
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