Predicting the styles that will prevail in a coming year is one of the most important and difficult problems in the fashion industry. A fashion designer must work on designs for the coming fall long before he or she can ?nd out for certain what styles are going to be "in." A well-known designer believes that there is a 0.20 chance that short dresses and skirts will be popular in the coming fall; a 0.35 chance that popular styles will be of medium length; and a 0.45 chance that long dresses and skirts will dominate fall fashions. The designer must now choose the styles on which to concentrate. If she chooses one style and another turns out to be more popular, pro?ts will be lower than if the new style were guessed correctly. The following table shows what the designer believes she would make, in hundreds of thousands of dollars, for any given combination of her choice of style and the one that prevails in the new season.
Prevailing Style
|
Designer's Choice
|
Short
|
Medium
|
Long
|
|
Short
|
8
|
3
|
1
|
|
Medium
|
1
|
9
|
2
|
|
Long
|
4
|
3
|
10
|
Construct a decision tree, and determine what style the designer should choose to maximize her expected pro?ts.