3.3 Two Basic Rules of Probability

Anna has to buy a new car. She has two choices, car A and car B. Anna can afford only one car. The probability that Anna will buy car A is $P \left(A\right) = 0 . 25$, and the probability that Anna will buy car B is $P \left(B\right) = 0 . 65$. Find:

• $P \left(A A N D B\right)$
• $P \left(A O R B\right)$

Carlos plays college soccer. He makes a goal 65% 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) = 0.65. B = the event Carlos is successful on his second attempt. P(B) = 0.65. Carlos tends to shoot in streaks. The probability that he makes the second goal $\left|\right.$ that he made the first goal is 0.90.

1. Wha…

Helen plays basketball. For free throws, she makes the shot 75% of the time. Helen must now attempt two free throws. C = the event that Helen makes the first shot. P(C) = 0.75. D = the event Helen makes the second shot. P(D) = 0.75. The probability that Helen makes the second free throw given that she made the first is 0.85. 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 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 play sports. Thirty of the seniors going directly to work play sports. Five of the seniors taking a gap year play sports. What is the probability that a senior is taking a gap year?

Let: M = math class, S = speech class, M$\left|\right.$S = math given speech

1. What is the probability that Felicity enrolls in math and speech?
   Find P(M $\cap$ S) = P(M$\left|\right.$S)P(S).
2. What is the probability that Felicity enrolls in math or speech classes?
   Find P(M $\cup$ S) = P(M) + P(S) - P(M $\cap$ S).
3. Are M and S independent? Is P(M$\left|\right.$S) = P(M)?
4. Are M and S mutually exclusive? Is P(M $\cap$ 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) = 0.40, P(D) = 0.30 and P(D$\left|\right.$B) = 0.5.

1. Find P(B $\cap$ D).
2. Find P(B $\cup$ D).

Studies show that about one woman in seven (approximately 14.3%) who live to be 90 will develop breast cancer. Suppose that of those women who develop breast cancer, a test is negative 2% of the time. Also suppose that in the general population of women, the test for breast cancer is negative about 85% of the time. Let B = woman develops breast cancer and let N = tests negative. Suppose one woman…

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 play sports. Thirty of the seniors going directly to work play sports. Five of the seniors taking a gap year play sports. What is the probability that a senior is going to college and plays sports?

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

1. Given that a woman develops breast cancer, what is the probability that she tests positive. Find P(P$\left|\right.$B) = 1 - P(N$\left|\right.$B).
2. What is the probability that a woman develops breast cancer and tests positive. Find P(B $\cap$ P) = P(P$\left|\right.$B)P(B).
3. What is the probability that a woman does not develop breast cancer. Find P(B′) = 1 - P(B).
4. What is the probabili…

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) = 0.40, P(D) = 0.30 and P(D$\left|\right.$B) = 0.5.

1. Find P(B′).
2. Find P(D $\cap$ B).
3. Find P(B$\left|\right.$D).
4. Find P(D $\cap$ B′).
5. Find P(D$\left|\right.$B′).

Suppose that one customer is randomly selected.

Find P(Z).

Suppose that one customer is randomly selected.

Find P(S).

Suppose that one customer is randomly selected.

Find P(S|Z).

Suppose that one customer is randomly selected.

In words, what is S|Z?

Suppose that one customer is randomly selected.

Find $P \left(Z \cap S\right)$.

Suppose that one customer is randomly selected.

In words, what is $Z \cap S$?

Suppose that one customer is randomly selected.

Are P and S independent events? Show why or why not.

Suppose that one customer is randomly selected.

Find $P \left(Z \cup S\right)$.

Suppose that one customer is randomly selected.

In words, what is $Z \cup S$?

Suppose that one customer is randomly selected.

Are Z and S mutually exclusive events? Show why or why not.

A survey was conducted in a large city to measure the popularity of that city’s mayor. The survey was repeated every year for three years. The survey polled 1,046 people to measure the mayor’s popularity. Everyone polled expressed either approval or disapproval. These are the results the poll produced:

• In Year 1, 60% of the population approved of the mayor’s actions in office.
• In Year 2, 57% of the…

Compute the probability of winning the following types of bets:

1. Betting on two lines that touch each other on the table as in 1-2-3-4-5-6
2. Betting on three numbers in a line, as in 1-2-3
3. Betting on one number
4. Betting on four numbers that touch each other to form a square, as in 10-11-13-14
5. Betting on two numbers that touch each other on the table, as in 10-11 or 10-13
6. Betting on 0-00-1-2-3
7. Betting…

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$\left|\right.$E) = _____
  4. P(G $\cap$ E) = _____
  5. P(G $\cup$ E) = _____
  6. Are G and E mutually exclus…

A special deck of cards has ten cards. Four are green, three are blue, and three are red. When a card is picked, its color of it 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, fol…

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 $\cup$ Z) = P(Y) + P(Z) - P(Y $\cap$ Z) using the information that Y and Z are independent events.
2. Use the rewritten rule to find P(Z) if P(Y $\cup$ Z) = 0.71 and P(Y) = 0.42.

According to the 2019 U.S. Census, approximately 331,449,281 people live in the United States. Of these people, 67,800,000 speak a language other than English at home. Of those who speak another language at home, 61.6% speak Spanish.

Let: E = speaks English at home; E' = speaks another language at home; 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 selfish than other people. They dropped 64 stamped, addressed envelopes with $10 cash in different classrooms on the George Washington campus. 44% were returned overall. From the economics classes 56% of the envelopes were returned. From the business, psychology, and history classes 31% were retu…

According to the American Red Cross, 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 45% of people have type O blood and 7% of people have Rh- factor; 38% of people have type O or Rh+ factor.

1. Find the probability that a person has both type O blood and the Rh- factor.
2. Find the probability that a person does NOT have…

In a box of assorted cookies, 36% contain chocolate and 12% contain nuts. In the box, 8% 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).