Question3. Each vertex of a triangle is colored white, black, green, blue, or orange. All configurations are equally probable. What is the probability that in a randomly chosen configuration there are exactly two vertices of the same color?
Studdy Solution
STEP 1
1. A triangle has 3 vertices.
2. Each vertex can be colored in 5 different colors: white, black, green, blue, or orange.
3. All color configurations are equally probable.
STEP 2
1. Calculate the total number of possible color configurations.
2. Calculate the number of configurations with exactly two vertices of the same color.
3. Determine the probability by dividing the number of favorable configurations by the total number of configurations.
STEP 3
Calculate the total number of possible color configurations. Each vertex can be colored in 5 different ways, so the total number of configurations is:
STEP 4
To have exactly two vertices of the same color, choose the color for the two vertices (5 ways), choose the two vertices to color the same (3 ways), and choose a different color for the third vertex (4 remaining colors).
Number of ways to choose the color for the two vertices:
Number of ways to choose which two vertices will have the same color:
Number of ways to choose a different color for the third vertex:
Total number of favorable configurations:
STEP 5
Determine the probability by dividing the number of favorable configurations by the total number of configurations:
The probability that there are exactly two vertices of the same color is:
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