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 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?

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.

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?

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

A silver dollar is flipped twice. Calculate the probability of each of the following occurring:

a) a head on the first flip

b) a tail on the second flip given that the first toss was a head

c) two tails

d) a tail on the first and a head on the second

e) a tail on the first and a head on the second or a head on the first and a tail on the second

f) at least one head on the two flips

In exercise 24 you calculated probabilities involving various blood types. Some of your answers depended on the assumption that the outcomes described were disjoint; that is, they could not both happen at the same time. Other answers depended on the assumption that the events were independent; that is, the occurrence of one of them doesn't affect the probability of the other. Do you understand the difference between disjoint and independent?

a) if you examine one person, are the events that the person is Type A and that the person is Type B disjoint or independent or neither?

b) If you examine two people, are the events that the first is Type A and the second Type B disjoint or independent or neither?

c) Can disjoint events ever be independent? Explain.

d) Write the definition of disjoint events and the definition of independent events. How do these definitions tell you the answers to parts a, b, and c?

You draw two cards from a standard deck of 52 cards, but before you draw the second card, you put the first on 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 3 in either order.

A local country club has a membership of 600 and operates facilities that include an 18-hole championship golf course and 12 tennis courts. Before deciding whether to accept new members, the club president would like to know how many members regularly use each facility. A survey of the membership indicates that 68% regularly use the golf course, 49% regularly use the tennis courts, and 5 % use neither of these facilities regularly. Find the probability that a randomly selected member uses the golf or tennis facilities regularly.

A sample is chosen randomly from a population that can be described by a Normal Model.
a) What's the sampling distribution model for the sample mean? Describe shape, center and spread.
b) if we choose a larger sample, what's the effect on this sampling distribution model?
c) Draw a sketch of the original population distribution and a sketch of the distribution of the samples (the sampling distribution)