3.3 Two Basic Rules of Probability

Carlos plays college soccer. He makes a goal 65 percent of the time he shoots. Carlos is going to attempt two goals in a row in the next game. A = the event Carlos is successful on his first attempt. P(A) = .65. B = the event Carlos is successful on his second attempt. P(B) = .65. Carlos tends to shoot in streaks. The probability that he makes the second goal given that he made the first goal is .…

Carlos plays college soccer. He makes a goal 65 percent of the time he shoots. Carlos is going to attempt two goals in a row in the next game. A = the event Carlos is successful on his first attempt. P(A) = .65. B = the event Carlos is successful on his second attempt. P(B) = .65. Carlos tends to shoot in streaks. The probability that he makes the second goal given that he made the first goal is .…

Carlos plays college soccer. He makes a goal 65 percent of the time he shoots. Carlos is going to attempt two goals in a row in the next game. A = the event Carlos is successful on his first attempt. P(A) = .65. B = the event Carlos is successful on his second attempt. P(B) = .65. Carlos tends to shoot in streaks. The probability that he makes the second goal given that he made the first goal is .…

Carlos plays college soccer. He makes a goal 65 percent of the time he shoots. Carlos is going to attempt two goals in a row in the next game. A = the event Carlos is successful on his first attempt. P(A) = .65. B = the event Carlos is successful on his second attempt. P(B) = .65. Carlos tends to shoot in streaks. The probability that he makes the second goal given that he made the first goal is .…

Helen plays basketball. For free throws, she makes the shot 75 percent of the time. Helen must now attempt two free throws. C = the event that Helen makes the first shot.
P(C) = .75. D = the event Helen makes the second shot. P(D) = .75. The probability that Helen makes the second free throw given that she made the first is .85. What is the probability that Helen makes both free throws?

A community swim team has 150 members. Seventy-five of the members are advanced swimmers. Forty-seven of the members are intermediate swimmers. The remainder are novice swimmers. Forty of the advanced swimmers practice four times a week. Thirty of the intermediate swimmers practice four times a week. Ten of the novice swimmers practice four times a week. Suppose one member of the swim team is chos…

A community swim team has 150 members. Seventy-five of the members are advanced swimmers. Forty-seven of the members are intermediate swimmers. The remainder are novice swimmers. Forty of the advanced swimmers practice four times a week. Thirty of the intermediate swimmers practice four times a week. Ten of the novice swimmers practice four times a week. Suppose one member of the swim team is chos…

A community swim team has 150 members. Seventy-five of the members are advanced swimmers. Forty-seven of the members are intermediate swimmers. The remainder are novice swimmers. Forty of the advanced swimmers practice four times a week. Thirty of the intermediate swimmers practice four times a week. Ten of the novice swimmers practice four times a week. Suppose one member of the swim team is chos…

A community swim team has 150 members. Seventy-five of the members are advanced swimmers. Forty-seven of the members are intermediate swimmers. The remainder are novice swimmers. Forty of the advanced swimmers practice four times a week. Thirty of the intermediate swimmers practice four times a week. Ten of the novice swimmers practice four times a week. Suppose one member of the swim team is chos…

A community swim team has 150 members. Seventy-five of the members are advanced swimmers. Forty-seven of the members are intermediate swimmers. The remainder are novice swimmers. Forty of the advanced swimmers practice four times a week. Thirty of the intermediate swimmers practice four times a week. Ten of the novice swimmers practice four times a week. Suppose one member of the swim team is chos…

A school has 200 seniors of whom 140 will be going to college next year. Forty will be going directly to work. The remainder are taking a gap year. Fifty of the seniors going to college are on their school's sports teams. Thirty of the seniors going directly to work are on their school's sports teams. Five of the seniors taking a gap year are on their schools sports teams. What is the probability…

Let M = math class, S = speech class, and M|S = math given speech.

1. What is the probability that Felicity enrolls in math and speech?
   Find P(M AND S) = P(M|S)P(S).
2. What is the probability that Felicity enrolls in math or speech classes?
   Find P(M OR S) = P(M) + P(S) − P(M AND S).
3. Are M and S independent? Is P(M|S) = P(M)?
4. Are M and S mutually exclusive? Is P(M AND S) = 0?

A student goes to the library. Let events B = the student checks out a book and D = the student checks out a DVD. Suppose that P(B) = .40, P(D) = .30, and P(D|B) = .5.

1. Find P(B AND D).
2. Find P(B OR D).

Researchers are studying one particular type of disease that affects women more often than men. Studies show that about one woman in seven (approximately 14.3 percent) who live to be 90 will develop the disease. Suppose that of those women who develop this disease, a test is negative 2 percent of the time. Also suppose that in the general population of women, the test for the disease is negative a…

Researchers are studying one particular type of disease that affects women more often than men. Studies show that about one woman in seven (approximately 14.3 percent) who live to be 90 will develop the disease. Suppose that of those women who develop this disease, a test is negative 2 percent of the time. Also suppose that in the general population of women, the test for the disease is negative a…

Researchers are studying one particular type of disease that affects women more often than men. Studies show that about one woman in seven (approximately 14.3 percent) who live to be 90 will develop the disease. Suppose that of those women who develop this disease, a test is negative 2 percent of the time. Also suppose that in the general population of women, the test for the disease is negative a…

Researchers are studying one particular type of disease that affects women more often than men. Studies show that about one woman in seven (approximately 14.3 percent) who live to be 90 will develop the disease. Suppose that of those women who develop this disease, a test is negative 2 percent of the time. Also suppose that in the general population of women, the test for the disease is negative a…

Researchers are studying one particular type of disease that affects women more often than men. Studies show that about one woman in seven (approximately 14.3 percent) who live to be 90 will develop the disease. Suppose that of those women who develop this disease, a test is negative 2 percent of the time. Also suppose that in the general population of women, the test for the disease is negative a…

Researchers are studying one particular type of disease that affects women more often than men. Studies show that about one woman in seven (approximately 14.3 percent) who live to be 90 will develop the disease. Suppose that of those women who develop this disease, a test is negative 2 percent of the time. Also suppose that in the general population of women, the test for the disease is negative a…

A school has 200 seniors of whom 140 will be going to college next year. Forty will be going directly to work. The remainder are taking a gap year. Fifty of the seniors going to college are on their school's sports teams. Thirty of the seniors going directly to work are on their school's sports teams. Five of the seniors taking a gap year are on their school's sports teams. What is the probability…

Refer to the information in the example. P = tests positive.

1. Given that a woman develops the disease, what is the probability that she tests positive? Find P(P|B) = 1 − P(N|B).
2. What is the probability that a woman develops the disease and tests positive? Find P(B AND P) = P(P|B)P(B).
3. What is the probability that a woman does not develop the disease? Find P(B′) = 1 − P(B).
4. What is the probability t…

A student goes to the library. Let events B = the student checks out a book and D = the student checks out a DVD. Suppose that P(B) = .40, P(D) = .30, and P(D|B) = .5.

1. Find P(B′).
2. Find P(D AND B).
3. Find P(B|D).
4. Find P(D AND B′).
5. Find P(D|B′).

Suppose that one citizen is randomly selected.

Find P(L).

Suppose that one citizen is randomly selected.

In words, what is C|L?

Suppose that one citizen is randomly selected.

In words, what is L AND C?

Suppose that one citizen is randomly selected.

Find P(L OR C).

Suppose that one citizen is randomly selected.

Are L and C mutually exclusive events? Show why or why not.

After a mayor of a major Canadian city announced his plans to cut budget costs in late 2011, researchers polled 1,046 people to measure the mayor’s popularity. Everyone polled expressed either approval or disapproval. These are the results their poll produced:

• In early 2011, 60 percent of the population approved of the mayor's actions in office.
• In mid-2011, 57 percent of the population approved of…

1. List the sample space of the possible items that are on the menu.
2. Find P(you will get a 17-oz. piece of meat).
3. Find P(you will get a pork chop).
4. Find P(you will get a 17-oz. pork chop).
5. Is getting a pork chop the complement of getting a chicken breast? Why?
6. Find two mutually exclusive events.
7. Are the events getting 17 oz. of meat and getting a pork chop independent?

Compute the probabilities.

1. P(you will get a chicken breast)
2. P(you will get a 17-oz. chicken breast)
3. P(you will get a chicken breast or you will not get a 17-oz. pork chop)
4. P(you will not get a chicken breast and you will get an 18-oz. pork chop)
5. P(you will get a piece of meat that is not 21 oz.)
6. P(you will get a piece of chicken that is not 21 oz.)
7. P(you will not get a chicken breast and you will…

Compute the probabilities:

1. P(you will not get a pork chop)
2. P(you will get a 20-oz. pork chop)
3. P(you will not get a chicken breast or you will not get an 18-oz. pork chop)
4. P(you will not get a chicken breast and you will not get an 18-oz. pork chop)
5. P(you will get a pork chop that is not 21 oz.)
6. P(you will not get a chicken breast or you will not get a pork chop)

Suppose that you have eight cards. Five are green and three are yellow. The five green cards are numbered 1, 2, 3, 4, and 5. The three yellow cards are numbered 1, 2, and 3. The cards are well shuffled. You randomly draw one card.

• G = card drawn is green
• E = card drawn is even-numbered
  1. List the sample space.
  2. P(G) = ________
  3. P(G|E) = ________
  4. P(G AND E) = ________
  5. P(G OR E) = ________
  6. Are G and E…

A special deck of cards has 10 cards. Four are green, three are blue, and three are red. When a card is picked, its color is recorded. An experiment consists of first picking a card and then tossing a coin.

1. List the sample space.
2. Let A be the event that a blue card is picked first, followed by landing a head on the coin toss. Find P(A).
3. Let B be the event that a red or green is picked, followed b…

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

1. List the sample space.
2. Let A be the event that there are at least two tails. Find P(A).
3. 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 one to three complete sentences, including justification.

Y and Z are independent events.

1. Rewrite the basic Addition Rule P(Y OR Z) = P(Y) + P(Z) - P(Y AND Z) using the information that Y and Z are independent events.
2. Use the rewritten rule to find P(Z) if P(Y OR Z) = .71 and P(Y) = .42.

Approximately 281,000,000 people over age five live in the United States. Of these people, 55,000,000 speak a language other than English at home. Of those who speak another language at home, 62.3 percent speak Spanish.

Let E = speaks English at home; E′ = speaks another language at home; and S = speaks Spanish.

Finish each probability statement by matching the correct answer.

Three professors at George Washington University did an experiment to determine if economists are more likely to return found money than other people. They dropped 64 stamped, addressed envelopes with $10 cash in different classrooms on the George Washington campus. Forty-four percent were returned overall. From the economics classes 56 percent of the envelopes were returned. From the business, p…

United Blood Services is a blood bank that serves more than 500 hospitals in 18 states. According to their website, a person with type O blood and a negative Rh factor (Rh–) can donate blood to any person with any bloodtype. Their data show that 43 percent of people have type O blood and 15 percent of people have Rh– factor; 52 percent of people have type O or Rh– factor.

1. Find the probability that…

In a box of assorted cookies, 36 percent contain chocolate and 12 percent contain nuts. Of those, 8 percent contain both chocolate and nuts. Sean is allergic to both chocolate and nuts.

1. Find the probability that a cookie contains chocolate or nuts (he can't eat it).
2. Find the probability that a cookie does not contain chocolate or nuts (he can eat it).