Probability (190 problems)

As she is leaving her home for class, a student grabs two pieces of fruit at random from her fruit bowl to take with her for an afternoon snack. In the fruit bowl are two oranges and two apples. Let x be the number of apples that the student chooses.

a) Draw a tree diagram to describe the 12 simple events in the above situation, the possible event outcomes, and the probability of each possible event outcome occurring (three decimal places is sufficient).

b) Create a table to display the probability distribution for p(x).

c) Draw a probability histogram for p(x).

d) What is the probability that the student chooses one or more apples in her selection of 2 pieces of fruit?

e) What is the mean value of x? State in a sentence what this mean value implies.

An eight-sided die, numbered 1-8, is rolled. Find the probability that the roll results in an even number or a number greater than six.

The plastic arrow on a spinner for a child’s game stops rotating to point at a color that will determine what happens next in the game. Determine whether the probability assignment is possible.

Probability of: red: 0.2; yellow, 0.2; green, 0.4; and blue, 0.2.

General: Valid Probabilities

(a) Explain why –0.41 cannot be the probability of some event.

(b) Explain Why 1.21 cannot be the probability of some event.

(c) Explain why 120% cannot be the probability of some event.

(d) Can the number 0.56 be the probability of an event? Explain.

In a large Introductory Statistics lecture hall, the professor reports that 55% of the students enrolled have never taken a Calculus course, 32% have taken only one semester of Calculus, and the rest have taken two or more semesters of Calculus. The professor randomly assigns students to groups of three to work on a project for the course. What is the probability that the first group mate you meet has studied

a) two or more semesters of Calculus?

b) some Calculus?

c) no more than one semester of Calculus?

d) Either no Calculus or two or more semesters of calculus.

What are the three self-evident truths about probabilities?

Many stores run "secret sales": Shoppers receive cards that determine how large a discount they get, but the percentage is revealed by scratching off that black stuff( what is that?) only after the purchase has been totaled at the register. The store is required to reveal (in the fine print) the distribution of discounts available. Which of these probability assignments are plausible?

(a) If you roll a single fair die and count the number of dots on top, what is the sample space of all possible outcomes? Are the outcomes equally likely?

(b) Assign probabilities to the outcomes of the sample space of part a .

(c) What is the probability of getting a number less than 5 on a single throw?

(d) What is the probability of getting 5 or 6 single throw?

A package of documents needs to be sent to a given destination, and delivery within one day is important. To maximize the chances of on-time delivery, three copies of the documents are sent via three different delivery services. Service A is known to have a 90% on-time delivery record, service B has an 88% on-time delivery record, and service C has a 91% on-time delivery record. What is the probability that at least one copy of the documents will arrive at its destination on time?

A Gallup Poll in June 2004 asked 1005 U. S. adults how likely they were read Bill Clinton's autobiography My Life. Here's how they responded:

If we select a person at random from this sample of 1005 adults,
a) what is the probability that the person responded "Will definitely not read it"?
b) what is the probability that the person will probably or definitely read it?
c) what is the probability that the person will probably or definitely not read it?

You roll 2 fair dice, a green one and a red one.

(a) Are the outcomes on the dice independent?

(b) Find P(1on green die and 2 on red die)

(c) Find P(2 on green die and 1 on red die)

(d) Find P(1 on green die and 2 on red die) or (2 on green die and 1 on red die)

In its monthly report, the local animal shelter states that it currently has 24 dogs and 18 cats available for adoption. Eight of the dogs and 6 of the cats are male. Find each of the following conditional probabilities if an animal is selected at random:

a) The pet is male, given that it is a cat.
b) The pet is a cat, given that it is a female.
c) The pet is female, given that it is a dog.
d) The pet is a dog, given that it is a male.

A local bank reports that 80 percent of its customers maintain a checking account, 60 percent have a savings account, and 50 percent have both. If a customer is chosen at random, what is the probability the customer has either a checking or a savings account? What is the probability the customer does not have either a checking or a savings account?

A device has three components and works as long as at least one of the components is functional. The reliabilities of the components are 0.96, 0.91, and 0.80. What is the probability that the device will work when needed?

General: Deck of Cards. You draw two cards from a standard deck of 52 cards, but before you draw the second card, you put the first one back and reshuffle the deck.

a.) Are the outcomes on the two cards independent? Why?

b.) Find P (3 on 1^st card and 10 on 2^nd).

c.) Find P (10 on 1^st card and 3 on 2^nd).

d.) Find the probability of drawing a 10 and a 3 in either order.

Agriculture: Cotton. A botanist has developed a new hybrid cotton plant that can withstand insects better than other cotton plants. However, there is some concern about the germination of seeds from the new plant. To estimate the probability that a seed from the new plan will germinate, a random sample of 3000 seeds was planted in warm, moist soil. Of these seeds, 2430 germinated.

a.) Use relative frequencies to estimate the probability that a seed will germinate. What is your estimate?

b.) Use relative frequencies to estimate the probability that a seed will not germinate. What is your estimate?

c.) Either a seed germinates, or it does not. What is the sample space in this problem? Do the probabilities assigned to the sample space add up to 1? Should they add up to 1? Explain.

d.) Are the outcomes in the sample space of part (c) equally likely?

According to exercise 4, the probability that a U. S. resident has traveled to Canada is 0.18, to Mexico is 0.09, and to both countries is 0.04.

a) What's the probability that someone who has traveled to Mexico has visited Canada too?
b) Are travel to Mexico and Canada disjoint events? Explain.
c) Are travel to Mexico and Canada independent events? Explain
d)What is the probability that someone who has traveled to Canada has also traveled to Mexico?

A sales representative must visit 4 cities: Omaha, Dallas, Wichita, and Oklahoma City. There are direct air connections between each of the cities. Use the multiplication rule of counting to determine the number of different choices the sales representative has for the order in which to visit the cities. How is this problem similar to problem 5?

Arches national park is located in southern Utah. The park is famous for it beautiful desert landscape and its many natural sandstone arches. Park ranger Edward Mccarrick started an inventory (not complete) of natural arches with the park that have an opening of at least 3 feet. The following table is based on information taken from the book canyon country arches and bridges, by F.A. Barnes. The height of the arch opening is rounded to the nearest foot.

For an arch chosen at random in arches national park, use the preceding information in estimate the probability that the height of the arch opening is

(b) 30 ft. or taller

(c) 3 to 49 ft.

(d) 10 to 74 ft.

(e) 75 feet or taller