Understanding the Probability Problem

We are asked to find the probability of drawing three balls of the same color in succession from a box containing balls of different colors. The key information is:

• Number of black balls = 2
• Number of yellow balls = 4
• Number of white balls = 6
• Total number of balls = \(2 + 4 + 6 = 12\)
• The balls are drawn with replacement. This means after each ball is drawn, it is put back into the box, so the total number of balls and the number of balls of each color remain constant for every draw.
• We need the probability that all three balls are the same color (all black OR all yellow OR all white).

Calculating Individual Probabilities

First, let's calculate the probability of drawing a ball of each specific color in a single draw:

• Probability of drawing a black ball (P(Black)) = \(\frac{\text{Number of black balls}}{\text{Total number of balls}} = \frac{2}{12} = \frac{1}{6}\)
• Probability of drawing a yellow ball (P(Yellow)) = \(\frac{\text{Number of yellow balls}}{\text{Total number of balls}} = \frac{4}{12} = \frac{1}{3}\)
• Probability of drawing a white ball (P(White)) = \(\frac{\text{Number of white balls}}{\text{Total number of balls}} = \frac{6}{12} = \frac{1}{2}\)

Calculating Probability of Same Color Draws

Since the draws are made with replacement, the probability of each draw is independent. We can calculate the probability of drawing three consecutive balls of the same color for each color:

• Probability of drawing 3 black balls (P(3 Black)) = P(Black) \(\times\) P(Black) \(\times\) P(Black) = \((\frac{1}{6})^3 = \frac{1}{216}\)
• Probability of drawing 3 yellow balls (P(3 Yellow)) = P(Yellow) \(\times\) P(Yellow) \(\times\) P(Yellow) = \((\frac{1}{3})^3 = \frac{1}{27}\)
• Probability of drawing 3 white balls (P(3 White)) = P(White) \(\times\) P(White) \(\times\) P(White) = \((\frac{1}{2})^3 = \frac{1}{8}\)

Combining Probabilities for the Final Answer

The event "all three balls are of the same color" can happen in three mutually exclusive ways: all three are black, OR all three are yellow, OR all three are white. To find the total probability, we add the probabilities of these three events:

Total Probability = P(3 Black) + P(3 Yellow) + P(3 White)

Total Probability = \(\frac{1}{216} + \frac{1}{27} + \frac{1}{8}\)

To add these fractions, we find a common denominator, which is 216:

• \(\frac{1}{27} = \frac{1 \times 8}{27 \times 8} = \frac{8}{216}\)
• \(\frac{1}{8} = \frac{1 \times 27}{8 \times 27} = \frac{27}{216}\)

Now, substitute these back into the sum:

Total Probability = \(\frac{1}{216} + \frac{8}{216} + \frac{27}{216} = \frac{1 + 8 + 27}{216} = \frac{36}{216}\)

Finally, simplify the fraction \(\frac{36}{216}\). Both the numerator and the denominator are divisible by 36:

Total Probability = \(\frac{36 \div 36}{216 \div 36} = \frac{1}{6}\)

Conclusion

The probability that all three balls drawn in succession with replacement are of the same color is \(\frac{1}{6}\).