Independent Events Example:

A bag contains 2 red marbles, 5 blue marbles, and 3 green marbles. A marble is selected and its color is noted. Then it is replaced. A second marble is selected and its color is noted. Find the probability of pulling two blues. 

Step 1:

We notice that since we replace the marble once we pull it the first time these are two independent events so we can use the formula

\[P(\text{Pulling Blue and Pulling Blue})=P(\text{Pulling Blue})\cdot P(\text{Pulling Blue})\]

Step 2: 

Now Calculate

\[P(\text{Pulling Blue})=\frac{5}{10}\]

so 

\[P(\text{Pulling Blue and Pulling Blue})=\frac{5}{10}\cdot\frac{5}{10}=\frac{1}{4}=0.25\]

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General Multiplication Rule Example:

A bag contains 2 red marbles, 5 blue marbles, and 3 green marbles. A marble is selected and its color is noted. Then it is NOT replaced. A second marble is selected and its color is noted. Find the probability of pulling two blues. 

Step 1:

We notice that since we do not replace the marble once we pull it the first time these are two dependent events so we can use the formula

\[P(\text{Pulling Blue and Pulling Blue})=P(\text{Pulling Blue})\cdot P(\text{Pulling Blue}|\text{Previously Pulled Blue})\]

Step 2: 

Now Calculate

\[P(\text{Pulling Blue})=\frac{5}{10}\]

and 

\[P(\text{Pulling Blue}|\text{Previously Pulled Blue}) =\frac{4}{9}\]

so 

\[P(\text{Pulling Blue and Pulling Blue})=\frac{5}{10}\cdot\frac{4}{9}=\frac{2}{9}\approx 0.22\]