In: Statistics and Probability
1. Survey results on the Age and Marital Status of women are given below. Use the data to answer the questions.
AGE
18 to 24 25 to 64 65 and over total
Married 3,046 48,116 7,767 58,929
Never Married 9,289 9,252 768 19,309
Widowed 19 2,425 8,636 11,080
Divorced 260 8,916 1,091 10,267
total 12,614 68,709 18,262 99,585
A. What is the probability of a randomly selected woman being Widowed?
B. What is the probability of a randomly selected woman being Divorced and 25 to 64?
C. What is the probability of a randomly selected woman being Divorced or 18 to 24?
D If a woman is 25 to 64 years of age, what is the probability that she is Widowed?
E. Are age and marital status independent events? Show your proof.
total number of women = total outcome = 99585
(A) favourable outcome = total number of widowed women = 11080 (number of widowed women)
total oucome = 99585
So, probability of a randomly selected woman being Widowed = (favourable outcome)/(total outcome) = 11080/99585
this gives, P(woman being widowed) = 0.1113
(B) favourable outcome = total number of woman being Divorced and 25 to 64, i.e. number of women who are divroced as well as in age group 25 to 64
using table, we get favourable outcome = 8916
total oucome = 99585
So, probability of a randomly selected woman being Divorced and 25 to 64= (favourable outcome)/(total outcome) = 8916/99585
this gives, P(woman being Divorced and 25 to 64) = 0.0895
(C) favourable outcome = total number of woman being Divorced or 18 to 24, i.e. number of women who are divorced or in age group 18 to 24. In this case, we will consider all women who are divorced and all women who are in age group of 18 to 24
using table, we get favourable outcome = 10267 + 12614 - 260 (260 is subtracted because it is counted twice, first in divorced women number and again in women in age group 18 to 24)
So, favourable outcome = 22621
total oucome = 99585
So, probability of a randomly selected woman being Divorced or 18 to 24= (favourable outcome)/(total outcome) = 22621/99585
this gives, P(woman being Divorced or 18 to 24) = 0.2272
(D) favourable outcome = number of women who are widowed in age category 25 to 64 = 2425
total outcome = total women in age group 25 to 64 = 68709
So, probability = (favourable outcome)/(total outcome) = 2425/68709
this gives, Probability = 0.0353
(E) We know that independent event follow the propoerty
let event A be that the selected woman is 18 to 24 and B be that the selected woman is married
then P(A) = favourable/total = (total number of women in 18 to 24)/(total number of women) = 12614/99585 = 0.1267
and P(B) = favourable/total = (total number of married women)/(total number of women) = 58929/99585 = 0.5918
and P(A and B) = (number of women who are married and in 18 to 24)/(total number) =3046/99585 = 0.0306
Now
P(A)*P(B) = 0.1267 * 0.5918 = 0.0750
and we have P(A and B) = 0.0306
So, it is clear that
Therefore, age and marital status are not independent events