A survey indicated that 65% of the families in an area have cars. Find, correct to three decimal places, the probability that among 7 families selected at r...
Question 1 Report
A survey indicated that 65% of the families in an area have cars. Find, correct to three decimal places, the probability that among 7 families selected at random in the area
(a) exactly 5 ;
(b) 3 or 4 ;
(c) at most 2 of them have cars.
Answer Details
This is a binomial situation with \(n=7\) families, probability of owning a car \(p=0.65,\) and \(q=1-p=0.35.\) The probability of exactly \(r\) successes is \(P(r)=\binom{7}{r}p^r q^{7-r}.\)
(a) Exactly 5 have cars.
\[P(5)=\binom{7}{5}(0.65)^5(0.35)^2=21\times0.116029\times0.1225\approx0.298.\]
(b) 3 or 4 have cars.
\[P(3)=\binom{7}{3}(0.65)^3(0.35)^4=35\times0.274625\times0.015006\approx0.144.\]
\[P(4)=\binom{7}{4}(0.65)^4(0.35)^3=35\times0.178506\times0.042875\approx0.268.\]
\[P(3\text{ or }4)=0.144+0.268\approx0.412.\]
(c) At most 2 have cars \((r=0,1,2).\)
\[P(0)=(0.35)^7\approx0.000643,\]
\[P(1)=\binom{7}{1}(0.65)(0.35)^6\approx0.008364,\]
\[P(2)=\binom{7}{2}(0.65)^2(0.35)^5\approx0.046590.\]
\[P(\text{at most }2)=0.000643+0.008364+0.046590\approx0.056.\]
Download The App On Google Playstore
Everything you need to excel in your exams
