Can I get help to solve the following problem

1.  It has been established by observing over a long period of time
that the probability a customer will buy apple is 0.4.  A customer
buys pear 50% of the time when apple is purchased, but only 10% of the
time when apple is not purchased.

a. construct a probabiity tree for apples and pears

with the help of probability tree, find the probability that a
customer will buy the following:
a. apple and pear
b. pear
c. apple or pear
d. pear but not apple

I would really appreciate your answer in step by step form so that I
can understand each step that should be taken
thanks
jabeda

Hi jabeda!!


When we have a sequences of outcomes, it is helpful to visualize the
outcomes by a tree diagram.

To draw a tree diagram follow these steps:
· Draw all the first-step outcomes at the first stage, these will be
the first stage vertices.
· Label the edges from the root to the first stage vertices with the
probabilities of the outcomes at the first stage.
· Draw the next level of the secuence, remember that each level of the
tree corresponds to one stage of the sequence.
· Label each vertex with one of the possible outcomes at this new
stage.
· Label each edge of this new stage with the probability of getting
the outcome at its right end given the sequence of outcomes that have
occurred so far.
· Repeat this process until you complete all the stages

Note that each path from the root to the far right of the tree diagram
represents a possible sequence of outcomes of our process. By the
definition of conditional probabilities, the probability of a path is
the product of the probabilities along its edges.

For this problem we have:

First stage:
An individual costumer can buy apples or not, so here we have all the
first-step outcomes at the first stage.

                                   |
                       0.4         |      0.6       
                ------------------------------------
                |                                  |             
              Apple                              No apple
                |                                  |
         0.5    |   0.5                      0.1   |   0.9
        -----------------                 -------------------
        |               |                 |                 |
       Pear          No pear             Pear            No pear    

        X               Y                 W                 Z

We have four path in the tree diagram: X, Y, W, Z , and each path has
its respective probability.

Now we can solve the problem:...find the probability that a customer
will buy:

a. apple and pear 
The path for apple AND pear is X, so we must calculate the probability
of X.
P(X) = 0.4 * 0.5 = 0.2

b. pear
The paths that result on buying pear are X and W, these are
independent events, then we must to calculate the probability of (X U
W).
P(X U W) = P(X) + P(W) = 0.4 * 0.5 + 0.6 * 0.1 = 0.2 + 0.06 = 0.26

c. apple or pear 
P(apple or pear) = P(apple) + P(pear) - P(apple and pear)
                 = 0.6 + 0.26 - 0.2 = 0.66

d. pear but not apple
The path for pear but not apple (or not apple and pear) is W, so we
must calculate:
P(W) = 0.6 * 0.1 = 0.06  


 I hope this helps you. Please feel free to request for an answer
clarification if you find something unclear here.

Best regards.
livioflores-ga

---

Clarification of Answer by livioflores-ga on 19 Aug 2003 23:45 PDT

I must clarify the "how to draw a tree diagram" ; I described a
horizontal (from left to right) tree and then draw a vertical (from up
to down) one, I did it for practical purposes, but also here is the
horizontal tree:
              
                 |- Pear     
              0.5|
       Apple ----| 
         |    0.5|  
      0.4|       |- No Pear
         | 
---------|
         |
      0.6|       |- Pear
         |    0.1|
      No apple --|
              0.9|
                 |- No pear   


Hope this clarify the answer.

Regards.
livioflores-ga

I thank you for your answer
jabeda