Exam 2 information
Material:
· Chapter 5: Sample spaces and events, probability, probability rules, disjoint events, conditional probability and independence, counting techniques including permutations and combinations and their application to probability, probability distributions for discrete random variables, expected value, binomial probability distribution.
In addition to being able to
do the basic statistical and probability computations, I suggest that you focus
on:
TI-83/84
calculators are recommended.
*You
may bring one 8.5” by 11” sheet of notes.
You may write on both sides. I
ask that you make up your own sheet and hand it in with your exam. Suggestions: include formulas, rules,
definitions, TI-83 keystrokes, etc.
I
encourage you to make up your sheet early and study by trying problems using
only the sheet.
Review problems:
1) According to a 2007 survey, 43% of all California teenagers own a surfboard and 38% of California teenagers owned a skateboard. The survey also showed that 28% of them owned both a surfboard AND a skateboard.
a. What is the probability that a teenager selected at random from California had either a surfboard or a skateboard (or both)? (hint: draw a Venn diagram)
b. What is the probability that a California teen selected at random has neither a surfboard nor a skateboard?
c. If we know that a California teen owns a surfboard, what is the probability he/she also owns a skateboard?
d. Are the events “owns a surfboard” and “owns a skateboard” disjoint? Why or why not?
e. Are the events “owns a surfboard” and “owns a skateboard” independent? Why or why not?
2) Consider the following game: an ordinary die is rolled once. If a one comes up, the player wins $35. If a 5 or 6 comes up, the player wins $4. If any other number comes up, the player wins nothing.
a. What is the probability of winning $4?
b. Suppose the game costs $5 to play. Let X = the player’s profit for one play of the game. Find the probability model for X (i.e. list all possible values of X along with the probabilities of each value.)
c. Find the expected value of the player’s profit.
d. Based on your answer to part c, would a player win or lose money in the long run? Explain.
3) Of the patients reporting to a clinic, 25% have strep throat, 50% have an allergy and 10% have both strep throat and an allergy.
a. What is the probability that a patient has an allergy GIVEN that they have strep throat?
b. What is the probability that a patient has at least one of these two ailments?
c. Are the events “has strep throat” and “has an allergy” independent? Why or why not? Justify your answer with probability.
4) How many ways can three outfielders and four infielders be chosen from five outfielders and seven infielders?
5) A quiz consists of six multiple-choice questions, each with three possible answer choices. How many different possible answer keys can be made?
6) One professor grades homework by randomly choosing 5 out of 12 homework problems to grade.
a. How many different groups of 5 problems are there from the 12 problems?
b. Silvia did 7 problems. What is the probability that one of the groups of 5 she completed comprised the group selected to be graded?
7) How many ways are there to arrange four pictures in a row on a wall?
8) Pyramid Lake is one of the best places in the country to catch trophy cutthroat trout. In this table x = number of fish caught in a 6-hour period. The percentage data are the percentages of fishermen who caught x fish in a 6-hour period while fishing from shore.
|
X |
0 |
1 |
2 |
3 |
4 or more |
|
% |
44% |
36% |
15% |
4% |
1% |
a. Find the probability that a fisherman selected at random fishing from shore catches two or more fish in a 6-hour period.
b. Compute m, the expected value of the number of fish caught per fisherman in a 6-hour period (round “4 or more” to 4).
9) Consider the experiment of rolling a pair of fair dice, say one green and one red, and observing the pair of numbers representing the number of dots on the upturned faces. The following are events from the sample space for the experiment: A = the green die shows an even number.
B = the sum of the dots on the two dice is
equal to 7.
a.
Find the
(theoretical) probabilities of each event.
b.
Are A
and B disjoint? Justify your answer
c.
Are A
and B independent? Justify your answer.
10) Below is the distribution of grades received on the 2002 AP Calculus exam
|
Grade x |
Proportion P(x) |
|
5 |
0.10 |
|
4 |
0.16 |
|
3 |
0.28 |
|
2 |
0.20 |
|
1 |
0.26 |
a.
What is
the probability that a score selected at random is at least 4?
b.
Find the
mean grade of this distribution.
c.
Plot a
histogram of the population and describe its shape.
11) Approximately 60% of all computer programmers are introverts. In a group of 5 computer programmers,
a. What is the probability that none are introverts?
b. What is the probability that 3 or more are introverts?
c. What is the probability that all 5 are introverts?
12) Past data suggests that 30% of all cars in a shopping mall parking lot belong to employees. Suppose 9 cars are selected at random from the lot.
a. What is the probability that none of the nine cars belong to employees?
b. What is the probability that at least one of the nine cars belongs to an employee? (Hint: use your answer to part a. as well as the complement of the event.
c. Find the mean and standard deviation of the number of cars that belong to employees.