1. Consider the experiment of tossing a fair coin three times. Please answer the following questions.

(a) List the elementary outcomes to write the sample space S of the experiment.

S={,,,,,,,}

(b) Suppose event A is "getting exactly one head," and event B is "getting a tail in the first flip". Please specify the two events as the sets of elementary outcomes, respectively. (E.g., A = {e1 , e2 , . . .} where ei 's are the actual outcomes in the above sample space.)

(c) Draw a Venn Diagram of the given experiment concerning events A and B.

(d) Based on the Venn Diagram, determine the following probabilities: P (A), P (B), P (AB), and P (A ? B).

Suppose a professor will select one day of the weekdays (Monday - Friday) for a make-up exam. Assume that Wednesday, Thursday, and Friday are equally likely; Monday and Tuesday are twice as likely as the former three days. Please answer the following questions about the modeling with probability.

(a) What are the probabilities of elementary outcomes?

(b) Suppose event A is "the make-up exam is on Tuesday or Wednesday". What is the probability of A?

(c) Suppose event B is "the make-up exam is on Friday". What is the probability of B (hat)?

Consider the two events A = [Obese] and B = [Male] for people in the age group 20-39 years old. A recent survey by the National Center for Health Statistics suggests the probabilities:

P(A) = 0.34; P(B) = 0.52; P(AB) = 0.15

for a randomly selected person. Please answer the following questions.

(Hint: Draw first a Venn Diagram with the probabilities for the following four regions: the region only for A, the region only for B, the region for AB, and the region for A ? B.)

(a) Determine the probabilities of the following events:

i ) A B (hat)

ii) A (hat) B (hat)

(b) Express the following events in the event notation and find their probabilities. i) Either A or B (hat) or both occur.

ii) Exactly one of the two events occurs.

4. An urn contains one green ball and four red balls.

(a) Suppose two balls will be drawn at random one after another and without replacement (i.e., the first ball drawn is not returned to the urn before the second one is drawn). Suppose we define two events as follows:

A = [Green ball appears in the first draw.] and B = [Green ball appears in the second draw.] i) Are the two events independent? Why or why not do you think so?

ii) Are the two events disjoint? Why or why not do you think so?

(b) Now, suppose two balls will be drawn with replacement (i.e., the first ball drawn will be returned to the urn before the second draw). Define two events A and B, in the same way in part (a).

i) Are the two events independent? Why or why not do you think so?

ii) Are the two events disjoint? Why or why not do you think so?

5. SupposeP(A)=0.23andP(B)=0.6.

(a) Determine P (A ? B) if A and B are disjoint.

(b) Determine P (A ? B) if A and B are independent.

Preview File Edit View Go Tools Window Help Q 35% Wed 6:58 PM MATH12_HW5.pdf (page 1 of 3) ~ ew Zoom Share Highlight Rotate Markup Search. MATH12_HW5.pdf 1. Consider the experiment of tossing a fair coin three times. Please answer the following questions. (a) List the elementary outcomes to write the sample space S of the experiment. S ={ (b) Suppose event A is "getting exactly one head," and event B is "getting a tail in the first flip". Please specify the two events as the sets of elementary outcomes, respectively. 1 (E.g., A = {e1, ez. ...} where e's are the actual outcomes in the above sample space.) (c) Draw a Venn Diagram of the given experiment concerning events A and B. 2 (d) Based on the Venn Diagram, determine the following probabilities: P(A), P(B), P(AB), and P(AU B). OCT 5 . ... 4Preview File Edit View Go Tools Window Help Q 35% Wed 6:58 PM MATH12_HW5.pdf (page 2 of 3) ~ ew Zoom Share Highlight Rotate Markup Search.. MATH12_HW5.pdf 2. Suppose a professor will select one day of the weekdays (Monday - Friday) for a make-up exam. Assume that Screen Shot Wednesday, Thursday, and Friday are equally likely; Monday and Tuesday are twice as likely as the former three 2022-10...8.05 PM days. Please answer the following questions about the modeling with probability. (a) What are the probabilities of elementary outcomes? 1 (b) Suppose event A is "the make-up exam is on Tuesday or Wednesday" . What is the probability of A? (c) Suppose event B is "the make-up exam is on Friday" . What is the probability of B? 3. Consider the two events A = [Obese] and B = [Male] for people in the age group 20-39 years old. A recent 2 survey by the National Center for Health Statistics suggests the probabilities: P(A) = 0.34; P(B) = 0.52; P(AB) = 0.15 for a randomly selected person. Please answer the following questions. (Hint: Draw first a Venn Diagram with the probabilities for the following four regions: the region only for A, the region only for B, the region for AB, and the region for A UB.) (a) Determine the probabilities of the following events: 1) AB OCT 5 4Preview File Edit View Go Tools Window Help Q 35% Wed 6:58 PM MATH12_HW5.pdf (page 2 of 3) ~ ew Zoom Share Highlight Rotate Markup Search.. MATH12_HW5.pdf 3. Consider the two events A = [Obese] and B = [Male] for people in the age group 20-39 years old. A recent survey by the National Center for Health Statistics suggests the probabilities: Screen Shot 2022-10...8.05 PM P(A) = 0.34; P(B) = 0.52; P(AB) = 0.15 for a randomly selected person. Please answer the following questions. (Hint: Draw first a Venn Diagram with the probabilities for the following four regions: the region only for A, the region only for B. the region for AB, and the region for A UB.) Screen Shot 1 (a) Determine the probabilities of the following events: 2022-10...8.14 PM i) AB ii) AB (b) Express the following events in the event notation and find their probabilities. i) Either A or B or both occur. 2 ii) Exactly one of the two events occurs. + OCT 5 . ...Preview File Edit View Go Tools Window Help Q 35% Wed 6:58 PM MATH12_HW5.pdf (page 3 of 3) ~ ew Zoom Share Highlight Rotate Markup Search. MATH12_HW5.pdf 4. An urn contains one green ball and four red balls. (a) Suppose two balls will be drawn at random one after another and without replacement (i.e., the first ball Screen Shot 2022-10...8.05 PM drawn is not returned to the um before the second one is drawn). Suppose we define two events as follows: 1 A = [Green ball appears in the first draw.] and B = [Green ball appears in the second draw.] i) Are the two events independent? Why or why not do you think so? Screen Shot 2022-10...8.14 PM ii) Are the two events disjoint? Why or why not do you think so? Screen Shot 2022-10...8.22 PM 2 (b) Now, suppose two balls will be drawn with replacement (i.e., the first ball drawn will be returned to the urn before the second draw). Define two events A and B, in the same way in part (a). i) Are the two events independent? Why or why not do you think so? ii) Are the two events disjoint? Why or why not do you think so? 3 OCT 5 . . . 4Preview File Edit View Go Tools Window Help Q 35% Wed 6:58 PM MATH12_HW5.pdf (page 3 of 3) ~ ew Zoom Share Highlight Rotate Markup Search. MATH12_HW5.pdf Screen Shot 2022-10...8.05 PM ii) Are the two events disjoint? Why or why not do you think so? 1 Screen Shot 2022-10...8.14 PM 5. Suppose P(A) = 0.23 and P(B) = 0.6. (a) Determine P(AUB) if A and B are disjoint. Screen Shot 2022-10...8.22 PM 2 (b) Determine P(AUB) if A and B are independent. Screen Shot 2022-10...8.29 PM Page 3 3 + OCT 5 . ... 4