Lecture 3 Flashcards
What is an Outcome?
The result of a single trial in a
probability experiment.
ex. Roll a dice
Outcome= 2
What is an Event?
Event: Consists of one or more outcomes and is a subset of the sample space.
ex.
Roll a 2…
or
Roll an even number
(2, 4, or 6)
What is Probability?
Probability: The likelihood of a stated outcome… “the probability of rolling a 2 for
a single die is 1/6”
Simple event
Event that consists a single outcome
ex. toss coin and roll die at same time
Simple event = tossing heads and rolling a 3, represented by H3.
ex.
Event A = {H3}
Not a simple event
Not a simple event = tossing heads and rolling an even number because there is more than one way to do that according to the
sample space.
❑Event B = {H2, H4, H6}
Simple event or not?
You roll a 6 sided die. The event you are looking at is rolling at least 4
Not simple
Simple or no?
Go around room and poll people’s ages. Then we pick one student to say their ages
Event that’s students age is between 18 and 23
NOt simple
Probability experiment
What is a sample space?
Probability experiment: An action, or trial,
through which specific results (counts,
measurements, or responses) are obtained.
Ex. roll a single die
The set of all possible outcomes of a probability experimnt
ex. 123456
Identifying the sample
What is the sample space and number of outcomes for a 1-question survey with
response options: yes, no, or unsure. You want
to divide responses by sex (M/F).
6 possible outcomes. Sample space is 6.
* M-Yes, M-No, M-Unsure, F-Yes, F-No, F-Unsure
Sample Space
❑ Sample Space: The set of all possible
outcomes of a probability experiment.
.
Roll a single die (dice)
{1,2,3,4,5,6}
You have a Likert question where one person
can respond Strongly Disagree, Disagree,
Agree, or Strongly Agree and you want to
examine responses in 4 age groups (18-24, 25-
34, 35-44, 45 +
16 possible outcomes. sample space is 16.
* SD-AGE1, SD-AGE2, SD-AGE3, SD-AGE4, D-
AGE1, D-AGE2, D-AGE3, D-AGE4…. And so
on…
Fundamental counting principle
The Fundamental Counting Principle = If one event can occur in m ways and a second event can occur in n ways, the number of ways the two events can occur in sequence is m ∙ n
ex. Can be extended to any # of events (e.g., m * n * o * p…)
-Hence blood types: 4 types x 2 Rh states = 8 possible outcomes
- 4 survey response options x 4 age groups = 16 possible outcomes
4-digit code
How many combinations?
Numbers that repeat or don’t repeat
1098*7= 5040 access codes (can’t repeate)
Can
10 options= 0,1,2,3,4, 5,6,7,8,9 (one for each #)
101010*10= 10000
Classical (theoretical) probability
In this form of probability equation – each outcome in the sample space has an equal likelihood of occurrence.
- This allows for some basic numerical operations on the
probabilities.
❑ Rolling a single die:
❑ Probability
(Event A = rolling a 3) = P(EA) = 1/6
❑ Probability
(Event B = rolling a 7) = 0 – trick question
(Theres No 7 on the die)
❑ Probability
(Event C = rolling < 5) = 4/6 = 2/3
.P(E) = Number of outcomes in event E
—————————————-
Number of outcomes in sample space
How to recognize something as conditional probability?
Key words: Given, if, assuming
Context: If you are asked to find the probability of one event, considering that another event has already happened or is known to occur, you are dealing with conditional probability. For example, “What is the probability of drawing a red card from a deck of cards, given that the card is a spade?”
What to do:
Mutually exclusive
Mutually exclusive = Two events A and B cannot occur at the same time because A and B have no outcomes in common
When
𝑃(𝐵∣𝐴)=𝑃(𝐵), it indicates that
𝑃(𝐵∣𝐴) means the “probability of event B occurring if we know that event A has already occurred.”
- conditional probability
P(B | A) = P(B) means that the chances of event B occurring will be the same whether or not event A has already occurred. This means that event B does not depend on event A occurring, and that is the definition of an event being independent. So, the statement P(B | A) = P(B) means that A and B are independent of each other.
In a genetics study of 7,324 pea plants, you observe a wrinkled pea seed 1,850 times. You then calculated the probability of getting a wrinkled seed to be 1,850/7,324 = 0.253. This is an example of which type of probability?
Empirical
Why:
The empirical probability approach involves observing some procedure and counting the number of times an event occurs.
Empiricle
Classical
Subjective
The empirical probability approach involves observing some procedure and counting the number of times an event occurs.
The classical probability approach involves finding how many ways an event can occur from a set of numbers of outcomes in the sample space.
The subjective probability approach involves estimating probability based on knowledge of the relevant circumstances.
How to recignize
Addition rule
multiplication rule conditional probability
This problem does not use the addition rule since we do not see the word “or.” This problem does not use the multiplication rule since we are only selecting one item. This problem is does use conditional probability since we do the word “given”.
In a shipment of 200 calculators, 3 are defective. What is the probability of selecting 2 calculators with one of them being defective and the other one being good.
Complement of event a is the
Probability that event a doesn’t happen
How many different ways can a teacher rearrange the students in the first row of their class that has 6 seats?
For the first desk in the row, there are 6 possibilities for whom the teacher could select to sit there. After the teacher selects the student for the first seat, they have 5 options for who to put in the second seat. Then, the teacher will have 4 options for the third seat and so on until the very last seat is opened. When it comes to the last seat, there will only be one student left to choose from. We can then use the fundamental counting principle to find the final answer for the total number of possibilities.
6×5×4×3×2×1 = 720
So, the teacher has 720 different possibilities for rearranging the students in the row.
Determine which numbers could be used to represent the probability of an event.
Question content area bottom
Part 1
Select all that apply.
A.
minus0.0009, because probability values must be between minus1 and 1.
B.
StartFraction 15 Over 15 EndFraction
, because probability values can equal 1.
Your answer is correct.C.
333.3%, because probability values can be greater than 100%.
D.
StartFraction 120 Over 101 EndFraction
, because probability values cannot be greater than 1.
E.
0, because probability values can equal 0.
Your answer is correct.F.
2.3, because probability values cannot be greater than 100.
0 and 1
When an event is almost certain to happen, its complement will be an unusual event.
Question content area bottom
Part 1
Determine the most appropriate conclusion.
A.
False, the complement of an event has an equal probability as the event.
B.
True, the complement would be an unusual event.
C.
False, the probability of the complement will have a higher probability than the event.
D.
False, the probability of the complement has no relation to the probability of the event.
b
8 of the 200 digital video recorders (DVRs) in an inventory are known to be defective. What is the probability you randomly select a DVR that is not defective?
0.96
Classify the following statement as an example of classical probability, empirical probability, or subjective probability. Explain your reasoning.
According to company records, the probability that a washing machine will need repairs during a nine-year period is 0.08.
empirical
he stated probability is calculated based on observations from the company records.
What is the difference between independent and dependent events?
Two events are independent when the occurrence of one event does not affect the probability of the occurrence of the other event. Two events are dependent when the occurrence of one event affects the probability of the occurrence of the other event
What does the notation P(B|A) mean?
The probability of event B occurring, given that event A has occurred.
Explain how the complement can be used to find the probability of getting at least one item of a particular type.
Question content area bottom
Part 1
Choose the correct answer below.
A.
The complement of “at least one” is “all.” So, the probability of getting at least one item is equal to P(all items)minus1.
B.
The complement of “at least one” is “all.” So, the probability of getting at least one item is equal to 1minusP(all items).
C.
The complement of “at least one” is “none.” So, the probability of getting at least one item is equal to P(none of the items)minus1.
D.
The complement of “at least one” is “none.” So, the probability of getting at least one item is equal to 1minusP(none of the items).
Getting “none of the items” is the set of all outcomes in the sample space that are not included in “at least one item.” Using the definition of the complement of an event and the fact that the sum of the probabilities of all outcomes is 1, the following formula is obtained.
P(at least one item)equals1minusP(none of the items)
d
By rewriting the formula for the Multiplication Rule, you can write a formula for finding
conditional probabilities. The conditional probability of event B occurring, given that event A has occurred, is Upper P left parenthesis Upper B vertical line Upper A right parenthesis equals StartFraction Upper P left parenthesis Upper A and Upper B right parenthesis Over Upper P left parenthesis Upper A right parenthesis EndFraction
. Use the information below to find the probability that a flight arrives on time given that it departed on time.
The probability that an airplane flight departs on time is 0.91.
The probability that a flight arrives on time is 0.87.
The probability that a flight departs and arrives on time is 0.82.
The probability that a flight arrives on time given that it departed on time is
0.901
If two events are mutually exclusive, why is Upper P left parenthesis Upper A and Upper B right parenthesis equals 0?
A.
Upper P left parenthesis Upper A and Upper B right parenthesis equals 0 because A and B are independent.
B.
Upper P left parenthesis Upper A and Upper B right parenthesis equals 0 because A and B are complements of each other.
C.
Upper P left parenthesis Upper A and Upper B right parenthesis equals 0 because A and B each have the same probability.
D.
Upper P left parenthesis Upper A and Upper B right parenthesis equals 0 because A and B cannot occur at the same time.
d
Can two events with nonzero probabilities be both independent and mutually exclusive?
Question content area bottom
Part 1
Choose the correct answer below.
A.
Yes, two events with nonzero probabilities can be both independent and mutually exclusive when their probabilities add up to one.
B.
No, two events with nonzero probabilities cannot be independent and mutually exclusive because independence is the complement of being mutually exclusive.
C.
Yes, two events with nonzero probabilities can be both independent and mutually exclusive when their probabilities are equal.
D.
No, two events with nonzero probabilities cannot be independent and mutually exclusive because if two events are mutually exclusive, then when one of them occurs, the probability of the other must be zero.
d
