Download Basic Concepts of Probability

Basic Concepts of Probability
Diana Pell
Definition 1. A sample space is a set of all possible outcomes of an experiment.
Exercise 1. An experiment consists of flipping a coin once. Find the sample space.
Exercise 2. An experiment consist of flipping a coin twice. Find the sample space.
Exercise 3. Find the sample space for the genders of three children in a family.
Exercise 4. An experiment consists of rolling a single die. Find the sample space.
Exercise 5. An experiment consists of rolling a green die and a red die. Find the sample space.
Exercise 6. Sample Space for the standard deck of 52 cards.
Figure 1:
Definition 2. An event is a subset of a sample space of an experiment.
Exercise 7. Suppose and experiment consists of tossing a coin three times and observing the
sequence of heads and tails. Determine the event E = “exactly two heads.”
Exercise 8. Suppose that we have two urns - call them urn I and urn II - each containing red
balls and white balls. An experiment consists of selecting an urn and then selecting a ball from
that urn and noting its color.
a) What is a suitable sample space for this experiment?
b) Describe the event “urn I is selected” as a subset of the sample space.
Definition 3. Experiments in which each outcome has the same probability are said to be
experiments with equally likely outcomes.
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Exercise 9. Experiment consists of flipping a coin two times. Find probability of every outcome
in the sample space.
Definition 4. If an experiment with sample space S has equally likely outcomes, then for any
event E the probability of E is given by
P (E) =
n(E)
n(S)
where n(E) and n(S) denote the number of elements in E and S, respectively.
Note: Probability is always a number from 0 to 1. Impossible events always have probability 0
and certain events have probability 1.
Exercise 10. A single die is rolled. Find the probability of getting
a) A 2.
b) A number less than 5.
c) An odd number.
d) A number less than four.
e) A number greater than seven.
Exercise 11. When two dice are rolled, find the probability of getting
a) A sum of 8.
b) Sum greater than or equal to 7.
c) Doubles (the same number on each die).
d) A sum less than 5.
Exercise 12. Three coins are flipped. Find the probability of getting
a) Exactly two tails.
b) At least two tails.
c) At most one tail.
Exercise 13. If a family has three children, find the probability that they have at least one boy
and one girl.
Exercise 14. A coin is flipped, and then a die is rolled. Use a tree diagram to find the probability
of getting heads on the coin and an even number on the die.
Exercise 15. Of the next 32 trials on the docket in a county court, 5 are homicides, 12 are drug
offenses, 6 are assaults, and 9 are property crimes. If jurors are assigned to trials randomly,
a) what’s the probability that a given juror won’t get a homicide case?
b) what’s the probability that a juror gets assigned to a case that isn’t a drug offense?
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Exercise 16. A 2-lb bag of Hershey’s miniatures contains 33 milk chocolate bars, 27 Krackel
bars, 26 Mr. Goodbars, and 19 Special Dark bars. Set up a frequency distribution and find the
probability that a bar chosen from a randomly selected bag is
a) A Mr. Goodbar.
b) A Krackel or Special Dark.
c) Not a milk chocolate bar.
Exercise 17. In a math class of seven women and nine men, if one person is selected at random
to come to the board to show the solution to a problem, what is the probability that the student
is a man?
Complement Rule:
P (E) = 1 − P (E 0 )
Definition 5. The complement of an event E is the set of outcomes in the sample space that
are not included in the outcomes of event E. The complement of E is denoted by E 0 (read “E
prime”).
Exercise 18. Find the complement of each event:
a) Selecting a month that has 31 days
b) Rolling two dice and getting a sum that is an odd number
Exercise 19. In a study, it was found that 23% of the people surveyed said that vanilla was
their favorite flavor of ice cream. If a person is selected at random, find the probability that the
person’s favorite flavor of ice cream is not vanilla.
Empirical Probability
The difference between classical and empirical probability is that classical probability assumes
that certain outcomes are equally likely (such as the outcomes when a die is rolled), while empirical probability relies on actual experience to determine the likelihood of outcomes.
Given a frequency distribution, the probability of an event being in a given class is
P (E) =
frequency for the class
f
=
total frequencies in the distribution
n
Exercise 20. In a random sample of 500 people, 210 had type O blood, 223 had type A, 51
had type B, and 16 had type AB. Set up a frequency distribution and find the probability that
a randomly selected person from the general population has
a) Type O blood.
b) Type A or B blood.
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c) Neither type A nor type O blood.
d) A blood type other than AB.
Exercise 21. In the travel survey below, find the probability that a person will travel by airplane over the Thanksgiving holiday.
Method
Frequency
Drive
41
Fly
6
Train or bus
3
Exercise 22. Hospital records indicated that knee replacement patients stayed in the hospital
for the number of days shown in the distribution.
Number of days stayed
Frequency
3
15
4
32
5
56
6
19
7
5
Find these probabilities.
a) A patient stayed exactly 5 days.
b) A patient stayed fewer than 6 days.
c) A patient stayed at most 4 days.
d) A patient stayed at least 5 days.
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