4.11 Probability topics: homework  (Page 2/6)

An experiment consists of tossing a nickel, a dime and a quarter. Of interest is the side the coin lands on.

• a List the sample space.
• b Let $A$ be the event that there are at least two tails. Find $\text{P(A)}$ .
• c Let $B$ be the event that the first and second tosses land on heads. Are the events $A$ and $B$ mutually exclusive? Explain your answer in 1 - 3 complete sentences, including justification.

Consider the following scenario:

• Let $\text{P(C)}=0.4$
• Let $\text{P(D)}=0.5$
• Let $\text{P(C|D)}=0.6$

• a Find $\text{P(C AND D)}$ .
• b Are $C$ and $D$ mutually exclusive? Why or why not?
• c Are $C$ and $D$ independent events? Why or why not?
• d Find $\text{P(C OR D)}$ .
• e Find $\text{P(D|C)}$ .

$E$ and $F$ mutually exclusive events. $P(E)=0\text{.}4$ ; $P(F)=0\text{.}5$ . Find $P(E\mid F)$ .

$J$ and $K$ are independent events. $\text{P(J | K)}=0.3$ .Find $P(J)$ .

$U$ and $V$ are mutually exclusive events. $P(U)=0\text{.}\text{26}$ ; $P(V)=0\text{.}\text{37}$ . Find:

• a $\text{P(U AND V)}$ =
• b $\text{P(U | V)}$ =
• c $\text{P(U OR V)}$ =

$Q$ and $R$ are independent events. $\mathrm{P(Q)}=0.4$ ; $\mathrm{P(Q AND R)}=0.1$ . Find $\mathrm{P(R)}$ .

$Y$ and $Z$ are independent events.

• a Rewrite the basic Addition Rule $\text{P(Y OR Z)}=\mathrm{P(Y)}+\mathrm{P(Z)}-\mathrm{P(Y AND Z)}$ using the information that Y and Z are independent events.
• b Use the rewritten rule to find $\mathrm{P(Z)}$ if $\mathrm{P(Y OR Z)}=0.71$ and $\mathrm{P(Y)}=0.42$ .

$G$ and $H$ are mutually exclusive events. $P(G)=0\text{.}5$ ; $P(H)=0\text{.}3$

• a Explain why the following statement MUST be false: $P(H\mid G)=0\text{.}4$ .
• b Find: $\text{P(H OR G)}$ .
• c Are $G$ and $H$ independent or dependent events? Explain in a complete sentence.

The following are real data from Santa Clara County, CA. As of a certain time, there had been a total of 3059 documented cases of AIDS in the county. They were grouped into the following categories ( Source: Santa Clara County Public H.D. ):

Suppose one of the persons with AIDS in Santa Clara County is randomly selected. Compute the following:

• a $\text{P(person is female)}$ =
• b $\text{P(person has a risk factor Heterosexual Contact)}$ =
• c $\text{P(person is female OR has a risk factor of IV Drug User)}$ =
• d $\text{P(person is female AND has a risk factor of Homosexual/Bisexual)}$ =
• e $\text{P(person is male AND has a risk factor of IV Drug User)}$ =
• f $\text{P(female GIVEN person got the disease from heterosexual contact)}$ =
• g Construct a Venn Diagram. Make one group females and the other group heterosexual contact.

Solve these questions using probability rules. Do NOT use the contingency table above. 3059 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 $\text{P(person is female) =}$
• b $\text{P(person obtained the disease through heterosexual contact) =}$
• c $\text{P(female GIVEN person got the disease from heterosexual contact) =}$
• d Construct a Venn Diagram. Make one group females and the other group heterosexual contact. Fill in all values as probabilities.