Chapter 3: Q.127 (page 232)

Use the following information to answer the next two exercises. The following are real data from Santa Clara County, CA. As of a certain time, there had been a total of $3, 059$ documented cases of AIDS in the county. They were grouped into the following categories:
Suppose a person with AIDS in Santa Clara County is randomly selected.
a. Find P(Person is female).
b. Find P(Person has a risk factor heterosexual contact).
c. Find P(Person is female OR has a risk factor of IV drug user).
d. Find P(Person is female AND has a risk factor of homosexual/bisexual).
e. Find P(Person is male AND has a risk factor of IV drug user).
f. Find P(Person is female GIVEN person got the disease from heterosexual contact).
g. Construct a Venn diagram. Make one group females and the other group heterosexual contact

Short Answer

Expert verified

a. $P (p e r s o n i s f e m a l e) = \frac{255}{3059}$

b. $P (p e r s o n h a s r i s k f a c t o r h e t r o s e x u a l c o n t a c t) = \frac{196}{3059}$

c. $P (P e r s o n i s f e m a l e O R h a s a r i s k f a c t o r o f I V d r u g u s e r) = \frac{788}{3059}$

d. $P (P e r s o n i s f e m a l e A N D h a s a r i s k f a c t o r o f h o m o s e x u a l / b i s e x u a l) = 0$

e. $P (P e r s o n i s m a l e A N D h a s a r i s k f a c t o r o f I V d r u g u s e r) = \frac{463}{3059}$

f. $P (P e r s o n i s f e m a l e G I V E N p e r s o n g o t t h e d i s e a s e f r o m h e t e r o s e x u a l c o n t a c t) = \frac{136}{196}$

g. The Venn diagram shown below:

Step by step solution

Content Introduction

The completed data in table is as follow:

Explanation (Part a)

Suppose a person with AIDS in Santa Clara County is randomly selected, therefore the probability of P(Person is female) is given below:

$P (p e r s o n i s f e m a l e s) = \frac{t o t a l f e m a l e s}{T o t a l i n d i v i d u a l s} P (p e r s o n i s f e m a l e s) = \frac{255}{3059}$

Explanation (Part b)

Suppose a person with AIDS in Santa Clara County is randomly selected, therefore the probability of P(Person has a risk factor heterosexual contact).

$P (P e r s o n h a s a r i s k f a c t o r h e t e r o s e x u a l c o n t a c t) = \frac{T o t a l p e r s o n w h o h a s a r i s k f a c t o r h e t e r o s e x u a l c o n t a c t}{t o t a l i n d i v i u d u a l s} P (P e r s o n h a s a r i s k f a c t o r h e t e r o s e x u a l c o n t a c t) = \frac{196}{3059}$

Explanation (Part c)

Suppose a person with AIDS in Santa Clara County is randomly selected, therefore the probability of P(Person is female OR has a risk factor of IV drug user).

$P (P e r s o n i s f e m a l e O R h a s a r i s k f a c t o r o f I V d r u g u s e r) = \frac{255}{3059} + \frac{533}{3059} P (P e r s o n i s f e m a l e O R h a s a r i s k f a c t o r o f I V d r u g u s e r) = \frac{788}{3059}$

Explanation (Part d)

Suppose a person with AIDS in Santa Clara County is randomly selected, therefore the probability of P(Person is female AND has a risk factor of homosexual/bisexual).

$P (P e r s o n i s f e m a l e A N D h a s a r i s k f a c t o r o f h o m o s e x u a l / b i s e x u a l) . = 0$

explanation (Part e)

Suppose a person with AIDS in Santa Clara County is randomly selected, therefore the probability of P(Person is male AND has a risk factor of IV drug user)

$P (P e r s o n i s m a l e A N D h a s a r i s k f a c t o r o f I V d r u g u s e r) = \frac{463}{3059}$

Explanation (Part f)

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

$P (P e r s o n i s f e m a l e G I V E N p e r s o n g o t t h e d i s e a s e f r o m h e t e r o s e x u a l c o n t a c t) = \frac{136}{196}$