Deck 28: Determining More Complicated Probabilities

Question 1

In tossing eight honest coins, HTHHTHTT is more probable than TTTTTTTT.

Answer 1

False

Question 2

If A and B are disjoint events, then P(A or B) = P(A) + P(B).

Answer 2

True

Question 3

In general for events A and B, P(A and B) = P(A) × P(B).

Answer 3

False

Question 4

In families of six children, the birth order BGGBGB is more probable than the birth order GGGBBB.

Question 5

A future birth order of BGBG has the same probability as does GGGG.

Question 6

Experiment: Draw a ball from Box 1, note its color, and then put the ball into Box 2. Then draw a ball from Box 2 and note its color.

A) Make a condensed tree diagram for this experiment, with probabilities on the branches.

B) Write the probabilities for each outcome of the experiment. (Write them on the tree diagram in part
C) For this experiment, what is the probability of the event "the second ball is B"?

Question 7

Three-step experiment: First, randomly choose one of the boxes described below. Then draw one marble. Without replacement, draw another marble from the same box. Note the colors of the two marbles (W = white, R = red).
Box 1 has four white marbles and two red marbles.
Box 2 has six white marbles and two red marbles.

A) Make a complete, condensed tree diagram for the experiment, including the outcomes and their probabilities.
B) What is the probability of at least one red marble?

Question 8

For some unusual two-step experiment, only the data below are given. What is P(XZ)? Write enough (numbers, calculations, words as needed) to make your thinking clear.

Question 9

For some unusual two-step experiment, only the data below are given. What is P(XZ)? Write enough (numbers, calculations, words as needed) to make your thinking clear.

Question 10

Below is a probability tree diagram for an experiment. If the probability of BD is 0.48, what is the probability of AC?

Question 11

Trisha tosses three fair pennies, but you can't see what happened. She tells you, however, "One or more of the pennies came up heads." What is the probability that Trisha tossed three heads? Write enough to make your thinking clear.

Question 12

Karen asked her dad for two dimes to buy gumballs for herself and her brother. (The machine gives one gumball for a dime.) There were three red gumballs, two yellow gumballs, and one green gumball in the machine. Her favorite gumball is red. She hopes that she will get at least one red gumball.

A) List all the possible outcomes she can get for the two gumballs (think carefully) and the probabilities for each.
B) What is the probability that Karen will get at least one red gumball?
C) What does your answer in part B mean?

Question 13

A) What does it mean to say that A and B are independent events?
B) Give an example of independent events.

Question 14

If you pick a book at random from the Washington School library, P(red cover) = 0.2 and P(fiction) = 0.4.

A) Find P(red cover and fiction). Show your work for credit.
B) What assumption did you make in part A? Justify your assumption for this situation.

Question 15

If you buy a blouse at random from the Sands End catalog, P(blue blouse) = 0.3 and P(frilly sleeves) = 0.2.

A) Find P(blue blouse or frilly sleeves). Show your work for credit.
B) What assumption did you make in part A? Justify your assumption for this situation.

Question 16

Experiment: Spin Spinner 1 (below) and note the number. Then spin Spinner 2 and note the number.

A) Make a tree diagram for this experiment, with all the final outcomes and the probabilities for each outcome.

B) P(the first spin is an even number) = ______
C) P(the sum of the two spins = 7) = ______
D) P(the first spin is an even number or the sum of the two spins = 7) = ______

Question 17

Experiment: Spin Spinner 1 (below) and note the number. Then spin Spinner 2 and note the number.

A) Make a tree diagram for this experiment, with probabilities on the branches of the tree.

B) Give the sample space for the experiment and the associated probabilities.
C) P(the first spin is an even number) = ______
D) P(the sum of the two spins = 9) = ______
E) P(the first spin is an even number or the sum of the two spins = 9) = ______

Question 18

A) Make a condensed tree diagram for this experiment. Spin and then select one ball from the box designated by the spinner. Record the outcomes and their probabilities.
B) What is the probability of drawing an R?

Question 19

Experiment: Spin the spinner below and note the number. Then spin the spinner again and note the number.

A) Give the sample space for the whole experiment.
B) P(first spin gives 3) = ______
C) What does your answer in part B mean? (If you can't answer part B, use a fake answer for it to answer part c.)
D) P(the sum of the two numbers = 6) = ______

Question 20

A box has 3 green balls (G) and 2 red balls (R). Experiment: Draw a ball and note its color. Do not put it back. Draw again and note the color.

A) Make a condensed tree diagram for the experiment, complete with probabilities for the outcomes.
B) What is the probability that the two balls drawn are the same color?

Question 21

Suppose that in your work you often deal with complicated events, as in the drawing below. (The circular regions show events.) Write a formula for how you could calculate P(A or B or C), using simpler probabilities like P(A) and so on.

Question 22

Suppose that in your work you often deal with complicated events as in the drawing below. Write a formula for how you could calculate P(A or B or C), using simpler probabilities like P(A) and so on.

Question 23

Give a formula for P(A or B or C) for events like those shown below, using simpler probabilities like P(A) and so on.

Question 24

For events A and B, if P(A) = 0.2 and P(B) = 0.4, then P(A or B) = 0.6.

Question 25

HHHHHHHHHHHH is impossible with 12 tosses of an honest coin.

Question 26

If events A and B are independent with P(A) = 0.4 and P(B) = 0.2, then P(A and B) = 0.8.

Question 27

If A and B are not disjoint events, then P(A or B) < P(A) + P(B).

Question 28

If A and B are independent events, P(A or B) = P(A) + P(B) - P(A) ° P(B).

Question 29

The probability of guessing correctly on all of five true-false questions is $\frac { 1 } { 5 }$

Question 30

Complete the following formulas for special cases, using the most concise answers. (Do not include any unnecessary probabilities.)

A) If A and B are disjoint events, then P(A or B) = _____.
B) If A and B are disjoint events, then P(A and B) = _____.

Question 31

In a certain game you toss two dice. If you roll a 6, you can start the game, and if you roll doubles (both dice with the same number of dots on top), you get an extra turn. Jody is hoping that he will roll a 6 and also get doubles on his next turn. Select the best answer for each item below and explain why you chose that response.

A) Rolling a 6 and rolling doubles are: INDEPENDENT NOT INDEPENDENT
B) Rolling a 6 and rolling doubles are: DISJOINT NOT DISJOINT

Question 32

A school tried out two curricula, P and Q, with a total of 200 children.

A) 60% of the 200 children studied curriculum P. Enter that number of children in the appropriate place in the contingency table.
B) 30% of all the children scored below average on the state test. Enter that number of children in the appropriate place in the table.
C) of the curriculum Q children scored below average. Enter that number of children in the appropriate place in the table.
D) Given that a child studied curriculum P, what is the probability that the child scored average or above average on the state test?
E) Given that child studied curriculum Q, what is the probability that the child scored average or above average on the state test?

Question 33

Two similar cities tried out two programs, P and Q, to cut down on the use of tobacco. They tested the programs with a total of 2000 junior high students.

A) 60% of the 2000 students were in program P. Enter that number of students in the appropriate place in the contingency table.
B) 30% of all the students used tobacco more. Enter that number of students in the appropriate place in the table.
C) of the program Q students used tobacco more. Enter that number of children in the appropriate place in the table.
D) Given that a student was in program P, what is the probability that the student used tobacco less or the same?
E) Given that a student was in program Q, what is the probability that the student used tobacco less or the same?

Question 34

In a polling of 400 parents, 20% were high income. One hundred and sixty low-income parents supported vouchers, and 50 high-income parents opposed vouchers. (Hint: Make a table before answering the questions.)

A) What is P(oppose vouchers | high income)? (That is, what is the probability that a parent opposes vouchers, given that the parent has high income?)
B) What is P(high income | oppose vouchers)?
C) What is the probability that a parent has high income OR opposes vouchers?

Question 35

Of 100 students randomly sampled, 40 are female and 60 are male. A fourth of the females say yes when asked if they have ever driven more than 90 miles/hour, the others say no. But two-thirds of the males say they have and only one-third say no. (Hint: Make a table before answering the questions.)

A) What is the probability that a student chosen at random will say yes to the question?
B) What is the probability that the student is female, given that the student says yes?
C) What is the probability that the student says yes and is female?
D) What is the probability that the student says yes or the student is female (or both)?

Question 36

A company has collected data on the numbers of male and female employees who did or did not graduate from college. The data are summarized below. (Hint: Make a table before answering the questions.)
If an employee is selected from this company, find the following probabilities.
If an employee is selected from this company, find the following probabilities.

A) P(female and college graduate)
B) P(female or college graduate)
C) P(female | college graduate)
D) P(college graduate | female)

Question 37

Here are data for one community for a year. (Hint: Make a table before answering the questions.)

A) Assuming that the data in the table are typical, what is the probability of a driver having one or more car crashes in a year?
B) Is the probability in part A theoretical or experimental? Explain briefly.
C) What is the probability that the driver is male, given that the driver is in a car crash?
D) What is the probability that the driver is in a car crash, given that the driver is male?
E) What is P(is in one or more car crashes OR is a male)?