1) C.M. Mooveys is planning to open a video rental store.   However,
the initial investment is $250,000.  He currently has this money in a
certificate of deposit earning 10%.  He may leave it there if he
decides not to open the store.  If he opens the store and it is
successful, he will generate a profit of $120,000.  If it not
successful, he will lose $150,000.  What would the probabilty of a
successful store have to be for C.M. to prefer this investing in a CD?

2)David N. Goliath is planning to open a sporting goods store. 
However, the initial investment is $100,000.  He currently has this
money in a cetificate of deposit earning 15%.  He may leave it there
if he decides not to open the store.  If he opens the store and it is
successful he will generate a profit of $40,000.  If it is not
successful, he will lose $80,000.  What would the probablity of a
succcessful store have to be for David to prefer this to investing in
a CD?

3)Before a marketing research study was done, John Colorado believed
there was a 50/50 chance that his music store would be a success.  The
research team determined that there is a 0.9 probabilty that the
marketing research will be favorable given a successful music store. 
There is also a 0.8 probabilty that the marketing research will be
unfavorable given an unsuccessful music store.

a)If the marketing research is favorable, what is the revised
probablity of a successful music store?
b)If the marketing research is unfavorable, what is the revised
probabilty of a successful music store?

Hi k9queen, 

In the order you posted them, here are your solutions:

1. 

If the payout from the CD is 10%, by doing nothing he earns $25,000. 
So, by opening the store he must get an expected profit of at least
$25,000 or he's better of not bothering.

Let x be the probability of success. Then:

x * 120,000 + (1-x) * -150,000 > 25,000 must be true.

120,000x + 150,000x - 150,000 > 25,000

270,000x > 175,000

x > 175,000/270,000

x > 35/54 = 0.6481

The probability of success must be at least 35/54 or 64.81% in order
for it to be a worthwhile endeavour.

----------

2. This question is basically the same as the first question, just
with different numbers.  The profit earned by doing nothing is 15% of
$100,000 or $15,000.  So the expected payout from opening the store
must be at least $15,000 in order for it to be worthwhile.

Let x = the probability of the store being successful.  Then:

40,000x + (1-x) * -80,000 > 15,000

40,000x + 80,000x - 80,000 > 15,000

120,000x - 80,000 > 15,000

120,000x > 95,000

x > 95/120 = 0.7916

So the probability of success must be 95/120 or 79.16% in order for
this to be a better option than the CD.

----------

3. For this I suggest you read the discussion about Bayes' Revised
Probability here: http://ubmail.ubalt.edu/~harsham/opre640a/partIX.htm#rbayapp

The prior reliability matrix is:

                         Market Research
Firm Success         Favorable         Unfavorable
  Success  (50%)        0.9               0.2      |  1.0
  Unsuccessful (50%)    0.1               0.8      |  1.0
                     -------------------------------------
                        1.0               1.0      |  2.0

The conditional probability matrix is then:

Firm Success         Favorable         Unfavorable
  Success  (50%)        0.45               0.1      |  0.55
  Unsuccessful (50%)    0.05               0.4      |  0.45
                     -------------------------------------
                        0.5                0.5      |  1.0

Normalizing the values (by dividing the cell values by the horizontal
sum in the outer column) produces the posterior probability matrix:

Firm Success         Favorable         Unfavorable
  Success  (50%)        0.81818            0.11111     |  1.0
  Unsuccessful (50%)    0.18182            0.88889     |  1.0
                     -------------------------------------
                        1.0                1.0         |  2.0

Drawing from that table we can now say that, with favorable market
research the revised probability of success is 81.81%, and the revised
probablity of success given unfavorable market research is 11.11%.

You can play around with larger matrices of values if you'd like by
using this excellent tool here:
http://ubmail.ubalt.edu/~harsham/Business-stat/matrix/matrix.htm

-----------------

I hope this helps you out and is clear enough to follow.

Hibiscus

Search Strategy: "revised probability", "finite math conditional probability"