Expert: Paul Klarreich - 11/24/2008

Question
Counting, Probability Distributions, and Further Topics in Probability

A car dealer has 6 red, 10 gray, and 7 blue cars in stock. Ten cars are randomly chosen to be displayed in front of the dealership. Find the probability that

a) 4 are red and the others are blue.
b) 3 are red, 3 are blue, and 4 are gray
c) exactly 5 are gray and none are blue
d) all 10 are gray

Answer
Questioner:   Jeffrey
Category:  Advanced Math
Private:  No
 
Subject:  Discrete Mathematics
Question:  Counting, Probability Distributions, and Further Topics in Probability

A car dealer has 6 red, 10 gray,

>> There is no such thing as a gray car.  Try buying one -- that car in the showroom is silver, please.

and 7 blue cars in stock. Ten cars are randomly chosen to be displayed in front of the dealership. Find the probability that

a) 4 are red and the others are blue.
b) 3 are red, 3 are blue, and 4 are gray
c) exactly 5 are gray and none are blue
d) all 10 are gray
..................................
Hi, Jeffrey,

The Basic Counting Principle says:

If A can be done NA ways and
If B can be done NB ways
then A and B together can be done NA*NB ways.

Now if there are 23 cars and you pick 10 of them, this can be done C(23,10) ways.  That will always be your denominator in:

a) 4 are red and the (6?) others are blue.

C(6,4) * C(7,6)

b) 3 are red, 3 are blue, and 4 are gray

C(6,3) * C(7,3) * C(10,4)

Get the idea, now?

c) exactly 5 are gray and none are blue

Guess what color the other five are.

d) all 10 are gray

How many possible choices are there?