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A bag contains red marbles, white marbles, and blue marbles. Randomly choose two marbles, one at a time, and without replacement. Find the following. Enter your answers as fractions or decimals rounded to three decimal places.
(a)The probability that the first marble is red and the second is blue.
P(first white and second blue) =
The probability that both are the same color.
P(same color) =

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MrRoyal

Answer:

[tex]P(First\ White\ and\ Second\ Blue) = \frac{3}{28}[/tex]

[tex]P(Same) = \frac{67}{210}[/tex]

Step-by-step explanation:

Given (Omitted from the question)

[tex]Red = 7[/tex]

[tex]White = 9[/tex]

[tex]Blue = 5[/tex]

Solving (a): [tex]P(First\ White\ and\ Second\ Blue)[/tex]

This is calculated using:

[tex]P(First\ White\ and\ Second\ Blue) = P(White) * P(Blue)[/tex]

[tex]P(First\ White\ and\ Second\ Blue) = \frac{n(White)}{Total} * \frac{n(Blue)}{Total - 1}[/tex]

We used Total - 1 because it is a probability without replacement

So, we have:

[tex]P(First\ White\ and\ Second\ Blue) = \frac{9}{21} * \frac{5}{21 - 1}[/tex]

[tex]P(First\ White\ and\ Second\ Blue) = \frac{9}{21} * \frac{5}{20}[/tex]

[tex]P(First\ White\ and\ Second\ Blue) = \frac{9*5}{21*20}[/tex]

[tex]P(First\ White\ and\ Second\ Blue) = \frac{45}{420}[/tex]

[tex]P(First\ White\ and\ Second\ Blue) = \frac{3}{28}[/tex]

Solving (b) [tex]P(Same)[/tex]

This is calculated as:

[tex]P(Same) = P(First\ Blue\ and Second\ Blue)\[/tex][tex]or\ P(First\ Red\ and Second\ Red)\ or\ P(First\ White\ and Second\ White)[/tex]

[tex]P(Same) = (\frac{n(Blue)}{Total} * \frac{n(Blue)-1}{Total-1})+(\frac{n(Red)}{Total} * \frac{n(Red)-1}{Total-1})+(\frac{n(White)}{Total} * \frac{n(White)-1}{Total-1})[/tex]

[tex]P(Same) = (\frac{5}{21} * \frac{4}{20})+(\frac{7}{21} * \frac{6}{20})+(\frac{9}{21} * \frac{8}{20})[/tex]

[tex]P(Same) = \frac{20}{420}+\frac{42}{420} +\frac{72}{420}[/tex]

[tex]P(Same) = \frac{20+42+72}{420}[/tex]

[tex]P(Same) = \frac{134}{420}[/tex]

[tex]P(Same) = \frac{67}{210}[/tex]

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