An urn contains 100 marbles: 20 white, 30 red, 50 green. Calculate the probability of selecting White, Red and Green marbles respectively. What is the probability of pulling a white, green, white and red marbles consecutively?

Apr 18, 2016

$\frac{950}{156849}$ or approximately 0.6%

Explanation:

Assuming the marbles are not replaced in the urn:

• The probability of the first marble being white is $\frac{20}{100}$

• The probability of the next marble being green is then $\frac{50}{99}$

• The probability of the next marble being white is $\frac{19}{98}$

• The probability of the next marble being red is $\frac{30}{97}$

So the probability of the sequence white, green, white, red is:

$\frac{20}{100} \cdot \frac{50}{99} \cdot \frac{19}{98} \cdot \frac{30}{97}$

$= \frac{10}{\textcolor{red}{\cancel{\textcolor{b l a c k}{50}}}} \cdot \frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{50}}}}{99} \cdot \frac{19}{98} \cdot \frac{30}{97}$

$= \frac{10 \cdot 19 \cdot 30}{99 \cdot 98 \cdot 97}$

$= \frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{3}}} \cdot 1900}{\textcolor{red}{\cancel{\textcolor{b l a c k}{3}}} \cdot 33 \cdot 98 \cdot 97}$

$= \frac{1900}{33 \cdot 98 \cdot 97}$

$= \frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{2}}} \cdot 950}{33 \cdot \textcolor{red}{\cancel{\textcolor{b l a c k}{2}}} \cdot 49 \cdot 97}$

$= \frac{950}{33 \cdot 49 \cdot 97}$

$= \frac{950}{156849} \approx 0.006$

That is approximately 0.6%

Aug 24, 2016

In support of Georg's solution

Explanation:

For probability questions of this type, if you are ever in doubt, draw a probability tree

$\textcolor{red}{\text{Assumption: this is selection without replacement}}$

From the diagram observe that the initial selection of
White ->20/100->20%

Red" "-> 30/100->30%

Green->50/100->50% From the probability tree the overall sequenced sampling probability of white: green: white: red is:

$\frac{20}{100} \times \frac{50}{99} \times \frac{19}{98} \times \frac{30}{97}$

For what follows refer to George's solution