A study found that 20% of people wanted no options on their cars, 60% wanted automatic transmissions, 50% wanted air conditioning and 30% wanted tape decks. In addition, they found that 25% wanted automatic transmissions and air conditioning. They found that 30% wanted automatic transmissions and tape decks, and 25% wanted air conditioning and tape decks. What is the probability that a customer will order all three?
Thanks guys
this is best done with a diagram like this one:
let "air conditioner" be A, "tape deck" be B, and "automatic transmission" be C. our strategy will be to give each area on the diagram a variable, then write equations for those variables, and finally solve the equations.
you're looking for the percent of people who want all three, which corresponds to the center of the diagram, so put the variable x there.
let's make more variables:
only A: a
A and B not C: b
only B: c
B and C not A: d
only C: e
A and C not B: f
the info in the problem gives:
a+b+f+x=60 (1)
b+c+d+x=50 (2)
d+e+f+x=30 (3)
b+x=25 (4)
d+x=30 (5)
f+x=25 (6)
a+b+c+d+e+f+x=80 (7)
this is a system of seven equations in seven unknowns. you could solve it with gaussian elimination, but it will be faster to do some substitutions.
a+f=35 (plug eqn 4 into eqn 1)
c+d=25 (4 into 2)
b+c=25 (5 into 2)
e+f=5 (5 into 3)
a+b=35 (6 into 1)
d+e=5 (6 into 3)
if I take these six equations and add them all up, I get
2(a+b+c+d+e+f)=130
a+b+c+d+e+f=65
plugging that into eqn 7
65+x=80
x=15
15% of people want all three
0% anyone wants a tape deck.
unless it's an 8 track -- then i can play my elvis tapes
