Question

question 12
of the 19 residents in an apartment complex, 7 own the units and the rest are renters. 8 residents are randomly interviewed regarding the living conditions at the apartment complex. what is the probability that of the 8 residents who were interviewed, 4 of them own the unit in which they live?
note: if your answer has up to four decimal places, please enter it below without rounding the number. but if your answer has more than four decimal places, round the number to four decimal places.

Explanation:

Step1: Calculate the number of renters

The total number of residents is \(n = 19\), the number of owners is \(k=7\), so the number of renters is \(19 - 7=12\).

Step2: Use the hyper - geometric probability formula

The hyper - geometric probability formula is \(P(X = x)=\frac{\binom{k}{x}\binom{N - k}{n - x}}{\binom{N}{n}}\), where \(N = 19\) (population size), \(k = 7\) (number of success states in the population), \(n = 8\) (number of draws), \(x = 4\) (number of observed successes).

• Calculate \(\binom{k}{x}=\binom{7}{4}=\frac{7!}{4!(7 - 4)!}=\frac{7\times6\times5\times4!}{4!\times3!}=\frac{7\times6\times5}{3\times2\times1}=35\)
• Calculate \(\binom{N - k}{n - x}=\binom{12}{4}=\frac{12!}{4!(12 - 4)!}=\frac{12\times11\times10\times9\times8!}{4!\times8!}=\frac{12\times11\times10\times9}{4\times3\times2\times1}=495\)
• Calculate \(\binom{N}{n}=\binom{19}{8}=\frac{19!}{8!(19 - 8)!}=\frac{19\times18\times17\times16\times15\times14\times13\times12\times11!}{8!\times11!}=\frac{19\times18\times17\times16\times15\times14\times13\times12}{8\times7\times6\times5\times4\times3\times2\times1}=75582\)

Step3: Calculate the probability

\(P(X = 4)=\frac{\binom{7}{4}\binom{12}{4}}{\binom{19}{8}}=\frac{35\times495}{75582}=\frac{17325}{75582}\approx0.2292\)

Answer:

\(0.2292\)