In a study of 67 movies made between 1982 and 1987, researchers found that the p
ID: 3329909 • Letter: I
Question
In a study of 67 movies made between 1982 and 1987, researchers found that the primary villain frequently wore at least one of the following clothing items: dark sunglasses, a trench coat, black boots. A more detailed reading of this research revealed some additional information:
• 44 villains wore dark sunglasses
• 21 villains wore a trench coat
• 11 villains wore black boots
• 13 villains wore dark sunglasses and a trench coat
• 4 villains wore dark sunglasses and black boots
• 2 villains wore a trench coat and black boots
• 1 villain wore dark sunglasses, a trench coat, and black boots
(a) If a villain is selected at random with each having equal probability, what is the probability of selecting a villain wearing at least one of the clothing items listed above?
(b) If a villain is selected at random with each having equal probability, what is the probability of selecting a villain wearing exactly one of the clothing items listed above?
Explanation / Answer
Total movies = 67
a) Number of villain who wear at least one of the clothing items = Number of villains wore dark sunglasses + Number of villains wore trench coat + Number of villains wore black boots - number of villains wore dark sunglasses and a trench coat - number of villains wore dark sunglasses and black boots - number of villains wore a trench coat and black boots + number of villain wore dark sunglasses, a trench coat, and black boots
= 44 + 21 + 11 - 13 - 4 - 2 + 1
= 58
P(selecting a villain wearing at least one of the clothing items listed above) = 58/67 = 0.8657
b) Number of villains wore dark sunglasses of villains wore dark sunglasses - number of villains wore dark sunglasses and a trench coat - number of villains wore dark sunglasses and black boots + number of villain wore dark sunglasses, a trench coat, and black boots
= 44 - 13 - 4 + 1
= 28
Number of villains wore trench coat of villains wore trench coat - number of villains wore dark sunglasses and a trench coat - number of villains wore trench coat and black boots + number of villain wore dark sunglasses, a trench coat, and black boots
= 21 - 13 - 2 + 1
= 7
Number of villains wore black boots of villains wore black boots - number of villains wore dark sunglasses and a black boots - number of villains wore trench coat and black boots + number of villain wore dark sunglasses, a trench coat, and black boots
= 11 - 4 - 2 + 1
= 6
probability of selecting a villain wearing exactly one of the clothing items listed above = (28 + 7 + 6) / 67 = 0.6119