3. Preliminary results from the U.S. National Health Interview Survey (July-Dece

Question

3. Preliminary results from the U.S. National Health Interview Survey (July-December 2016) indicate that 50.8% of American households do not have a landline but do have at least one wireless phone; a PDF of the article is at https://www.ede.gov/nc /w 1705.pdf. Suppose you choose 4 U.S. households at random. Here, because 4 is such a small fraction of the over 125 million households in the U.S. (Source: www.statistica.com ) we will assume that households are independent. Let W- "household does not have a landline but does have at least one wireless phone". For events (a) through (d) below list the simple events and calculate the probability of the event. a) All four households do not own a landline but do have at least one wireless phone? b) None of the four households do not own a landline but do have at least one wireless phone? c) Exactly one of the four households do not own a landline but do have at least one wireless phone? d) At least one of the four households do not own a landline but do have at least one wireless phone? (Only calculate the probability here; do not list the simple events because there are too many).

Explanation / Answer

#3.
p = 0.508

As per binomial distribution,
P(X=r) = nCr * p^r * (1-p)^(n-r)

a)
P(X = 4) = 4C4*0.508^4*(1-0.508)^0
P(X = 4) = 0.0666

b)
P(X=0) = 4C0 * 0.508^0 * (1-0.508)^4
P(X = 0) = 0.0586

c)
P(X = 1) = 4C1 * 0.508^1 * (1-0.508)^3
P(X = 1) = 0.2420

d)
P(X >= 1) = 1 - P(X=0)
= 1 - 0.0586
= 0.9414

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