Question

Twenty-six percent of couples who plan to marry this year are planning destination weddings. Assume the variable is binomial. In a random sample of 11 couples who plan to marry, find the probability of the following. Round the answers to four decimal places.

Source: Time magazine.

Part 1

At least 6 couples will have a destination wedding

P (at least 6 couples will have a destination wedding) =.0412

Part 2 out of 3

Fewer than 5 couples will have a destination wedding

P (fewer than 5 couples will have a destination wedding) =

Answer 1

Given

p = 0.26      ( 26% of couple are planning destinatoion wedding )

n = 11           ( We have random sample of 11 couples )

We are assuming that variable is Binomial distribution .

i.e X ~ B ( n =11 , p = 0.26 )

Its PDF will be given by

P(X=x) = $\binom{n}{x} p^x (1-p)^{n-x}$                                ..... (**)

(a)

To find Probability that atleast 6 couples will have a destinatoion wedding .

So we need to find P( X $\geq$ 6 )

P( X $\geq$ 6 ) = P(X=6) + P(X=7) + P(X=8) + P(X=9) + P(X=10) + P(X=11)

    = $\binom{11}{6} 0.26 ^x (1-0.26 )^{11-6}$ + $\binom{11}{7} 0.26 ^x (1-0.26 )^{11-7}$ +......+$\binom{11}{11} 0.26 ^x (1-0.26 )^{11-11}$

{ Using ( ** ) }

After calculation we get

P( X $\geq$ 6 ) = P(X=6) + P(X=7) + P(X=8) + P(X=9) + P(X=10) + P(X=11)

P( X $\geq$ 6 ) = 3.16695*10^-02 +7.94793*10^-03 +1.39626*10^-03 +1.63526*10^-04 +1.1491*10^-05 + 3.67034*10^-07

P( X $\geq$ 6 ) = 0.04118907 $\approx$ 0.0412

Hence

Pr( atleast 6 couples will have a destinatoion wedding ) = 0.0412

(b)

To find Probability that fewer than 5 couples will have a destinatoion wedding .

So here we need to find P( X < 5 )

P( X < 5 ) = P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4)

                 =

    = $\binom{11}{0} 0.26 ^x (1-0.26 )^{11-0}$ + $\binom{11}{1} 0.26 ^x (1-0.26 )^{11-1}$ +$\binom{11}{2} 0.26 ^x (1-0.26 )^{11-2}$+......+$\binom{11}{4} 0.26 ^x (1-0.26 )^{11-4}$

    = 0.03643753 + 0.14082613 + 0.24739725 + 0.26077007 + 0.18324383

P( X < 5 ) = 0.8686748 $\approx$ 0.8687

Hence

P( fewer than 5 couples will have a destinatoion wedding ) = 0.8687

{ Note that :- In part (b) i.e Part 2 of 3 , if we were asked that to find probability almost 5 couples will have a destinatoion wedding then we would be finding P(X $\leq$ 5 ) , but in this case , since we need to find probability of fewer than 5 , so we find P( X < 5 )   }