4.6 Exercises  (Page 2/2)

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 $\text{P(C AND D)}$ .
• 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 March 31, 2000, there was 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 $\text{P(female GIVEN person got the disease from heterosexual contact)}$ =

A previous year, the weights of the members of the San Francisco 49ers and the Dallas Cowboys were published in the San Jose Mercury News . The factual data are compiled into the following table.

For the following, suppose that you randomly select one player from the 49ers or Cowboys.

• a Find the probability that his shirt number is from 1 to 33.
• b Find the probability that he weighs at most 210 pounds.
• c Find the probability that his shirt number is from 1 to 33 AND he weighs at most 210 pounds.
• d Find the probability that his shirt number is from 1 to 33 OR he weighs at most 210 pounds.
• e Find the probability that his shirt number is from 1 to 33 GIVEN that he weighs at most 210 pounds.
• f If having a shirt number from 1 to 33 and weighing at most 210 pounds were independent events, then what should be true about $\text{P(Shirt\# 1-33 | ≤ 210 pounds)}$ ?

Approximately 249,000,000 people live in the United States. Of these people, 31,800,000 speak a language other than English at home. Of those who speak another language at home, over 50 percent speak Spanish. ( Source: U.S. Bureau of the Census, 1990 Census )

Let: $E$ = speak English at home; $\mathrm{E\text{'}}$ = speak another language at home; $S$ = speak Spanish at home

Finish each probability statement by matching the correct answer.

Find $\text{P(hit was made by Babe Ruth)}$ .

• A $\frac{\text{1518}}{\text{2873}}$
• B $\frac{\text{2873}}{\text{12351}}$
• C $\frac{\text{583}}{\text{12351}}$
• D $\frac{\text{4189}}{\text{12351}}$

Find $\text{P(hit was made by Ty Cobb | The hit was a Home Run)}$

• A $\frac{\text{4189}}{\text{12351}}$
• B $\frac{\text{1141}}{\text{1720}}$
• C $\frac{\text{1720}}{\text{4189}}$
• D $\frac{\text{114}}{\text{12351}}$

Are $\text{the hit being made by Hank Aaron}$ and $\text{the hit being a double}$ independent events?

• A Yes, because $\text{P(hit by Hank Aaron | hit is a double) = P(hit by Hank Aaron)}$
• B No, because $\text{P(hit by Hank Aaron | hit is a double) ≠ P(hit is a double)}$
• C No, because $\text{P(hit is by Hank Aaron | hit is a double) ≠ P(hit by Hank Aaron)}$
• D Yes, because $\text{P(hit is by Hank Aaron | hit is a double) = P(hit is a double)}$