Chapter 3: Q.128 (page 232)

Answer these questions using probability rules. Do NOT use the contingency table. Three thousand fifty-nine cases of AIDS had been reported in Santa Clara County, CA, through a certain date. Those cases will be our population. Of those cases, $6.4 %$ obtained the disease through heterosexual contact and $7.4 %$ are female. Out of the females with the disease, $53.3 %$ got the disease from heterosexual contact.
a. Find P(Person is female).
b. Find P(Person obtained the disease through heterosexual contact).
c. Find P(Person is female GIVEN person got the disease from heterosexual contact)
d. Construct a Venn diagram representing this situation. Make one group females and the other group heterosexual contact. Fill in all values as probabilities.

Short Answer

Expert verified

a. $P (P e r s o n i s f e m a l e) = 0.074$

b. $P (P e r s o n o b t a i n e d t h e d i s e a s e t h r o u g h h e t e r o s e x u a l c o n t a c t) = 0.064$

c. $P (P e r s o n i s f e m a l e G I V E N p e r s o n g o t t h e d i s e a s e f r o m h e t e r o s e x u a l c o n t a c t) = 0.62$

d. Venn diagram so formed is:

Step by step solution

Content Introduction

We are given,

Three thousand fifty-nine cases of AIDS had been reported in Santa Clara County, CA, through a certain date.

Explanation (Part a)

from the problem it is given that $3059$ individuals in which $7.4 %$ is female. If person of AIDS has been reported in Santa Clara County, CA, then the probability of the person is female is

$P (P e r s o n i s f e m a l e) = 0.074$

Explanation (part b)

from the problem it is given the $3059$ individuals in which $6.4 %$ obtained disease through heterosexual contact than the probability is that

$P (P e r s o n o b t a i n e d t h e d i s e a s e t h r o u g h h e t e r o s e x u a l c o n t a c t) = 0.064$

Explanation (part c)

Suppose a person with AIDS in Santa Clara County is randomly selected probability P(Person is female GIVEN person got the disease from heterosexual contact) is given by,

$P (P e r s o n i s f e m a l e G I V E N p e r s o n g o t t h e d i s e a s e f r o m h e t e r o s e x u a l c o n t a c t) = \frac{P (P e r s o n i s f e m a l a n d g o t t h e d i e s e a s e f r o m h e t r o s e x u a l c o n t a c t)}{P (p e r s o n g o t t h e d i s e a s e f r o m h e t r o s e x u a l c o n t a c t)} P (P e r s o n i s f e m a l e G I V E N p e r s o n g o t t h e d i s e a s e f r o m h e t e r o s e x u a l c o n t a c t) = \frac{0.074 \times 0.533}{0.064} P (P e r s o n i s f e m a l e G I V E N p e r s o n g o t t h e d i s e a s e f r o m h e t e r o s e x u a l c o n t a c t) = 0.62$