According to the “season’s cleaning” article, the U.S. Department of Energy repo

Question

According to the “season’s cleaning” article, the U.S. Department of Energy reports that 25% of people with two- cars garages don’t have room to park any cars inside. (January 1, 2009 Rochester D&C) Assuming this to be true, what is the probability of the following?
a) Exactly 3 two-car garage household of a random sample of 5 two-car garage households do not have room to park any cars inside.
b) Exactly 7 two-car garage households of a random sample of 15 two-car garage households do not have room to park any cars inside.
c) Exactly 20 two-car garage households of a random sample of 30 two-car garage households do not have room to park any cars inside.

Explanation / Answer

This is binomial probability. If there is a probability p of an event happening, the probability that that event happens exactly r out of n times is

P = nCr (p)r (1 - p)n-r.

Think of having n slots for an event to happen -- we want the event to happen in exactly r of them (each time with probability p) and the event *not* to happen in exactly n-r of them (each time with probability 1-p). If we take one of these configurations of the event happening r times and multiply all the probabilities, that gives (p)r (1 - p)n-r. We then multiply this by the number of possible configurations that give the even happening exactly r times, which is nCr. If you need more help on this formula let me know.

For all of these problems, p = 25% = 1/4.

a) n = 5, r = 3

P = 5C3 (1/4)3 (3/4)2

(Notice -- we are looking for a configuration in which 3 of the households meet an event with a probability of 1/4 (no parking room) and 2 of them meet an event with a probability of 3/4 (yes parking room), and then multiplying by the number of ways that is possible.)

P = 10 (1/64) (9/16)

P = 90/1024

P = 45/512.

b) n = 15, r = 7

P = 15C7 (1/4)7 (3/4)8

P = 6435 (1/16384) (6561/65536)

P = .03932.

c) n = 30, r = 20

P = 30C20 (1/4)20 (3/4)10

These numbers are huge so I am doing the whole thing on the calculator and just writing the answer:

P = 1.5388 x 10-6

or a probability of about one one-millionth.

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