Suppose that a basket with 3 balls: gold, silver, and copper. We are going to select three balls with replacement(independent).

Question

(a) Draw a tree diagram;
(b) Determine the probability distribution;
(c) Verify the outcome by the binomial probability formula.

Answer 1

OK, picture time:

 

       G (GGG selected)

   G  S (GGS selected)  

       C (GGC selected)

       G (GSG selected)

G S  S (GSS selected)

       C (GSC selected)

       G (GSG selected)

   C  S (GCS selected)

       C (GCC selected)

       

The above represents (horizontally) the first third of the tree...with the outcomes listed in parenthesis...

For example (GCC selected) means that the first ball we got was Gold, the Second Copper and the third Copper.

Redrawing this same tree two more times starting with a first selection of Silver and Copper (instead of Gold) will complete the diagram!

 

Assuming the order of selection is irrelevant, how many times is each 3 ball combinations selected?

Count up the leaves of the tree for each of the possible labels we have, this gives a PDF for this selection process.

 

Note there is only 1 way to select GGG, SSS, and CCC...but there are multiple ways to select 1 ball of each color...namely GSC, GCS, SCG, SGC, CGS, and CSG...6 total ways (3! many). Divide this count by the total number of outcomes...number of leaves in the tree...this should be 3^3 since we have 3 choices for each independent selection!

 

Verification amounts to checking that your probabilities match up with the binomial formula.

 

Hope this helps!