Transcript Probability - DePaul QRC

Probability
Three Types of Probabilities
• Theoretical
• Relative Frequency
• Personal or Subjective
Determining the Probability of an Outcome
Theoretical Probability
Example: assume coins made such that they are equally likely to
land with heads or tails up when flipped.
Probability of a flipped coin showing heads up is ½.
Number of desired outcomes/ total number of possible
outcomes
Relative Frequency
Example: observe the number of male and female births in a
given city over the course of a year.
In 1987 there were a total of 3,809,394 live births in the
U.S., of which 1,951,153 were males.
Probability of male birth is 1,951,153/3,809,394 = 0.5122
The Relative-Frequency
Relative-frequency interpretation: applies to situations that
can be repeated over and over again and observed.
Examples:
• Buying a weekly lottery ticket and observing whether
it is a winner.
• Testing individuals in a population and observing
whether they carry a gene for a certain disease.
• Observing births and noting if baby is male or female.
Long-Run Relative Frequency
Probability =
proportion of time an outcome occurs over the long run
Long-run relative frequency of males born in the United States is
about 0.512.
Possible results for relative frequency of male births:
Proportion of male births jumps around at first but
starts to settle down just above .512 in the long run.
Tossing a coin n times….
The proportion of heads in “n” tosses of a coin changes as we make
more tosses. Eventually it approaches 0.5
Summary of Relative-Frequency
Interpretation of Probability
• Can be applied when situation can be repeated
numerous times and outcome observed each time.
• Relative frequency should settle down to constant
value over long run, which is the probability.
• Does not apply to situations where outcome one time
is influenced by or influences outcome the next time.
• Cannot be used to determine whether outcome will
occur on a single occasion but can be used to predict
long-term proportion of times the outcome will occur.
Personal Probability
Personal probability: the degree to which a given
individual believes the event will happen. Can’t be
tested through repetition.
The probability must be between 0 and 1 and be
coherent.
Examples:
• Probability of finding a parking space
downtown on Saturday.
• Probability that a particular candidate for a
position would fit the job best.
Computing theoretical probabilities
To calculate the probability of an event, if every outcome is equally likely.
1. Count all the possible outcomes of the random process.
2. Count the outcomes that are favorable to that event
The probability is calculated as the ratio
probability=
# favorable outcomes
----------------------------# all possible outcomes
Example: One deck of cards is shuffled and the top card is placed face down on
the table. What is the chance that the card is a king of hearts?
1.
2.
How many cards are in a deck?
How many king of hearts?
Chance=1/52
Computing theoretical probabilities
Suppose there are M possible outcomes for one process and N
possible outcomes for a second process. The total number of
possible outcomes is for the two processes combined is M x N.
Examples:
•How many outcomes are possible when you roll two dice?
•A restaurant menu offers two choices for an appetizer, five
choices for a main course, and three choices for a dessert. How
many different three-course meals?
•A college offers 12 natural science classes, 15 social science
classes, 10 English classes, and 8 fine arts classes. How many
choices do you have for a schedule comprised of one of each
class?
Creating and Using a Sample Space
The sample space is the list of all possible outcomes for an event (or
events).
For example:
If recording gender, the sample space is:
{male, female}
If rolling a die:
{1, 2, 3, 4, 5, 6}
If rolling a die twice:
{(1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (2,1)…(6,4) (6,5) (6,6)}
--Notice that (1,2) and (2,1) are separate outcomes in the sample space.
Order matters!
Creating and Using a Sample Space
Let’s take a look at the sample space for a family of three:
{BBB, BBG, BGB, GBB, GGB, GBG, BGG, GGG}
What is the probability of a family with three children having only
one boy?
{GGB, GBG, BGG}  3/8
What is the probability that the second child in a family with three
children will be a boy?
{BBB, BBG, GBB, GBG}  4/8 = ½
What is the probability that a family with three children will have
only one boy who is also the second child?
{GBG}  1/8
Applying Some Simple
Probability Rules
The sum of the probabilities of all possible
outcomes must be one. If A, B, and C are the
only possible outcomes:
pr(A) + pr(B) + pr(C) = 1
Example: If probability of a single
birth resulting in a boy is 0.512, then
the probability of it resulting in a girl
is 0.488.
Simple Probability Rules
If two outcomes cannot happen simultaneously,
they are said to be mutually exclusive.
The probability of one or the other of two
mutually exclusive outcomes happening is the
sum of their individual probabilities.
pr(A or B) = pr(A) + pr(B)
Example: If you think probability of getting
an A in your statistics class is 50% and
probability of getting a B is 30%, then
probability of getting either an A or a B is
80%. Thus, probability of getting C or less is
20%.
Simple Probability Rules
If two outcomes could happen simultaneously, they are
not mutually exclusive. The probability of one or the
other of two mutually exclusive outcomes happening
is the sum of their individual probabilities minus the
probability that they both happen.
pr(A or B) = pr(A) + pr(B) – pr(A and B)
Example: What is the probability of getting a number that
is even or a multiple of 3 on one roll of a die?
The probability of rolling an even number is 3/6. The
probability of rolling a multiple of 3 is 2/6. But the
probability of rolling a number that is both even and a
multiple of 3 is 1/6. So the probability of rolling either a
even number or a multiple of 3 is 3/6 + 2/6 – 1/6 = 4/6
Simple Probability Rules
If two events do not influence each other, the
events are said to be independent of each other.
If two events are independent, the probability
that they both happen is found by multiplying
their individual probabilities.
pr(A and B) = pr(A)×pr(B)
Example: A woman will have two children.
Assume the probability the birth results in boy is
0.512. Then probability of having 2 boys in a row is
0.512×0.512 = 0.262144. About a 26% chance a
woman will have 2 boys.
Simple Probability Rules
The probability of getting at least one of a possible
outcome out of numerous trials is equal to 1 minus
the probability of not getting that outcome at all.
pr(at least one) = 1 – pr(none)
Example: If a die is rolled twice, what is the
probability that at least one of the rolls is a 2?
The probability of not rolling a 2 both times =
5/6 × 5/6 = 23/36
1- 25/36 = 11/36
Simple Probability Rules
Example: You purchase 10 lottery tickets, for which
the probability of winning some prize is 1 in 10. What
is the probability that you will have at least one
winning ticket?
Probability of winning something is 1/10 = 0.1
Probability of winning nothing is 9/10 = 0.9
P( at least one winner in 10 tickets) =
1 – P( losing 10 times in a row) =
1 – (.9)*(.9)*(.9)*(.9)*(.9)*(.9)*(.9)*(.9)*(.9)*(.9)
1 - 0.910 = 0.651
Simple Probability Rules
Another Example: What is the probability that a
region will experience at least one 100-year flood
during the next 100 years?
Probability of a flood is 1/100.
Probability of no flood is 99/100.
P( at least one flood in 100 years) =
1 - .99100 = 0.634
Answer the following questions. Determine
whether each is asking for a theoretical,
relative frequency, or personal probability?
If you flip a coin and do it fairly, what is the
probability that it will land heads up?
What is the probability that you will
eventually own a home; that is, how likely
do you think it is?
A bag contains some marbles. You pull one
marble 10 times and get a red marble 6
times. What is the probability of getting a
red marble?
Explain what’s wrong with this partial
answer to one of the previous questions:
“The probability that I will eventually
own a home is 1/2 because either it will
happen or it won’t.”