Question

A theme park conducts a study of families that visit the park during a year. The fraction of such families of size m is 8−m28,m=1,2,3,4,5,6, and 7.For a family of size m that visits the park, the number of members of the family that ride the roller coaster follows a discrete uniform distribution on the set {1, ... , m}.Calculate the probability that a family visiting the park has exactly six members, given that exactly five members of the family ride the roller coaster.

Answer 1

Answer:

Step-by-step explanation:

The probability of famy size $M=m$ is

$P(M=m)=\frac{8-m}{28},m=1,2,...7$

Let $N$ members of family ride the roller coaster, then

$P(N=n|M=m)=\frac{1}{m},n=1,2,...m=0$

otherwise

Required probability=

$P(M=6|N=5)=\frac{P(N=5,M=6)}{P(N=5)}\\\\=\frac{P(N=5|M=6)P(M=6)}{P(N=5, M=5)+P(N=5, M=6)+P(N=5, M=7)}$

(Since $P(N=n|M=m)=0$ if n>m $P(N=n,M=m)=0$

$\frac{P(N=5|M=6)P(M=6)}{P(N=5|M=5)P(M=5)+P(N=5|M=6)P(M=5)+P(N=5|M=7)P(M=7)}\\\\=\frac{\frac{1}{6}\times \frac{2}{28}}{\frac{1}{5}\times \frac{3}{28}\times\frac{1}{6}\times\frac{2}{28}\frac{1}{7}\times\frac{1}{28}}=0.3097$