After testing asphalt shingles are classified as high medium

After testing, asphalt shingles are classified as high, medium, or low grade, and different grade shingles are sold under different warranties.

(a) In how many ways can the next 15 shingles be classified into high, medium, or low grade? (Hint. Think of a task consisting of 15 stages, with each stage having 3 outcomes.

(b) How many classifications have three, five, and seven shingles in the high, medium, and low grade categories, respectively?

(c) If the classifications are all equally likely, what is the probability of the event in part (b)?

Solution

(a)

Of the next 15 shingles the first one may high,medium or low, the second one also high, medium or low and so on.

Total number of ways = 3*3* .... * 3 (15 times) = 3¹⁵ =14348907

(b)

There should be 3 high grade , 5 medium grade and 7 low grade shingles which can be done in (¹⁵C₃¹²C₅⁷C₇ ) = 360360.

(c)

The probability in part b is = 360360/14348907 = 0.0251 (approx.) by classical definition.