Question

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.

Answer 1

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