Answer to Question #261959 in Statistics and Probability for shri

Question

dem0graphic profile of ecuator in 2018

27.08% population are 0-14 years old

18.35% population are 15-24 years old

39.59 population are 25-54 years old

7.53% population are 55-64 years old

7.45% population are 65 years old

number of random sample of 20 people .what is probability that 2 are 65 years ols, 5 are 55-64 years old, 6 are 25-54 years old ,the number of random sample that have composition like that one described above is________.Each of those random sample have a probability of occuring of _________the probability of event containing all random samples that fit the description above is______________

options

a) 97772875200

b)1.1649

c)0.0001138

d)349186

e)0.319

Answer 1

To solve this question, we shall apply the multinomial distribution as described below. The multinomial distribution states that,

Let $c$ denote the number of outcomes categories with their probabilities as, $\pi_1,\pi_2,...,\pi_c$

where $\displaystyle\sum_{i=1}^c(\pi_i)=1.0$ then for $n$ independent observations, the multinomial probabilities that $n_1$ falls in category 1, $n_2$ falls in category 2, ...., $n_c$ falls in category c is given by

$p(n_1,n_2,..,n_c)=n!/(n_1!n_2!..n_c!)*\pi_1^{n_1}\pi_2^{n_2}*...*\pi_c^{n_c}$

For this question, $n=20,\pi_1=0.2708, \pi_2=0.1835,\pi_3=0.3959,\pi_4=0.0753,\pi_5=0.0745$

and $n_3=6,n_4=5,n_5=2$

Therefore, for this case, $n=6+5+2=13$ and the probability is,

$p(n_3=6,n_4=5,n_5=2)=13!/(6!5!2!)(0.3959^6)*(0.0753^5)*(0.0745^2)=1.407929e+13* 5.17369e-11=1.864391e-06$

Therefore, the probability that 2 are 65 years old, 5 are 55-64 years old, 6 are 25-54 years old is

The number of random sample that have composition like that one described above is 5, each of those random sample have a probability of occurring of $\pi_i$, the probability of event containing all random samples that fit the description above is 1.0.