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Step-by-Step Solution
Step 1: Understand the Problem
We have a bag of six balls, each of a different color. We draw balls with replacement.
• First, we draw two balls in succession and want the probability (denoted by p ) that both are of the same color.
• Next, we draw four more balls in succession and want the probability (denoted by q ) that exactly three of them are of the same color.
Finally, we are given that the ratio p : q = m : n where m and n are coprime, and we wish to find m + n .
Step 2: Compute p (Probability that Both Balls are the Same Color)
When the first ball is drawn, it can be any of the six colors. Since the drawing is with replacement, the second ball has the same 6 -color choice.
• The chance that the second ball is the same color as the first is 1/6 .
Therefore,
p = \frac{1}{6}.
Step 3: Compute q (Probability that Exactly Three Out of Four Draws are the Same Color)
We draw four balls with replacement. We want exactly three to be of the same color, while the fourth is a different color.
• Total number of possible color sequences for four draws is 6^4 = 1296 .
• To count the favorable outcomes:
1. Choose the color that appears exactly three times: ^6C_1 = 6 ways.
2. Choose the different color for the remaining ball: ^5C_1 = 5 ways (because it must differ from the chosen triple color).
3. Decide in which one of the four positions the different color appears: there are 4!/3! = 4 ways.
Thus, the number of favorable outcomes is 6 \times 5 \times 4 = 120 .
Hence,
q = \frac{120}{6^4} = \frac{120}{1296} = \frac{5}{54}.
Step 4: Find the Ratio p : q
We have
p = \frac{1}{6},
\quad
q = \frac{5}{54}.
Computing the ratio:
\frac{p}{q}
= \frac{\frac{1}{6}}{\frac{5}{54}}
= \frac{1}{6} \times \frac{54}{5}
= \frac{54}{30}
= \frac{9}{5}.
Therefore, p : q = \frac{9}{5} , which means m = 9 and n = 5 .
Step 5: Compute m + n
Since m = 9 and n = 5 ,
m + n = 9 + 5 = 14.
Final Answer
The value of m + n is 14 .
