Chapter 4
Posterior probability
A revised probability based on additional information.
Event
A subset of a sample space.
What is the intersection symbol for A and B and what does it mean?
A ∩ B ; A and B
Probabilities are always between ___ and ___
0-1
Let A and B be two independent events with P(A) = 0.40 and P(B) = 0.20. Which of the following is correct? P(B|A)=0.40𝑃(𝐵|𝐴)=0.40 P(A|B)=0.08𝑃(𝐴|𝐵)=0.08 P(A∩B)=0𝑃(𝐴∩𝐵)=0 P(A∪B)=0.52
0.52 P(A U B)= P(A) + P(B) - P(A n B)
The likelihood of Company A's stock price rising is 15%, and the likelihood of Company B's stock price rising is 80%. Assume that the returns of Company A and Company B stock are independent of each other. The probability that the stock price of at least one of the companies will rise is __________.
0.83 https://imgur.com/a/jWPJtVN
The sum of the probabilities of all possible outcomes is...
1
What is the union symbol for A and B and what does it mean?
A ∪ B ; A or B ; "OR", "At least"
Joint Probabilities are the probabilities of the ____ of 2 events
intersection
Anthony feels that he has a 75% chance of getting an A in statistics and a 55% chance of getting an A in Managerial Economics. He also believes that he has a 40% chance of getting an A in both classes. What is the probability that he doesn't receive an A in either class?
Apply the Addition Rule (previous flashcard) to get .90 for getting an A in either class. Then subtract 1 (all probabilities equal 1) by 0.90 to get 0.10.
What is the complement of A?
Aᶜ
Which of the following represents an empirical probability? The probability of tossing a head on a coin is 0.5. The probability of rolling a 2 on a single die is one in six. A skier believes she has a 0.10 chance of winning a gold medal. Based on past observation, a manager believes there is a three-in-five chance of retaining an employee for at least one year.
Based on past observation, a manager believes there is a three-in-five chance of retaining an employee for at least one year.
When updating a prior probability based on new information, which of the following methodologies is MOST useful? Bayes' Theorem The Central Limit Theorem The Empirical Rule Chebyshev's Theorem
Bayes' Theorem
Used when each outcome in a sample space is equally likely to occur
Classical probability
Prior probability
Reflects what we know now before the arrival of any new information
An experiment consists of tossing three fair coins. What is the probability of observing two tails?
3/8 Explanation: https://imgur.com/a/E4YRDfX
Probability calculated as a relative frequency of occurrence
Empirical probability
Process that leads to one of several possible outcomes
Experiment
Conditional probability formula
Joint probability / Marginal Probability Ex: For those under 30, the conditional probability of enrolling is 0.06 / 0.27 = 0.22 https://imgur.com/ByrtZJy Another example: https://imgur.com/a/9HsE6Ve
Simple Event and example
Just one of the possible outcomes of an experiment ; getting an A in a course
Formulas: If A and B are independent events
P (A | B) = P(A) ; P(A n B) = P (A | B) x P(B) = P(A) x P(B)
A and B are mutually exclusive when
P(A n B) = 0
Conditional Probability Formula ; Probability that A occurs given B occurs ;
P(A | B ) = P(A ∩ B) / P(B)
What represents the probability of A and B if the B is a conditional?
P(A | B) ; the vertical line stands for given
The Multiplication Rule Formula and when to use it ;
P(A ∩ B) = P(A | B ) x P(B) ; the joint probability of events A AND B
Solve for P (A ∪ B)
P(A) + P(B) - P(A ∩ B) Reason: If we simply sum P(A) and P(B) then we overstate the probability because we double count the probability of the intersection of A and B. This is why P(A ∩ B) needs to be subtracted from P(A) + P(B)
Bayes' theorem formula for the probability of A given B
P(A|B) = P(B|A)P(A)/P(B)
The complement rule
P(Aᶜ) = 1 - P(A)
Probability calculated by drawing on personal and subjective judgement
Subjective probability
Bayes' theorem is calculated by using what rule in the denominator? The multiplication rule. The addition rule. The total probability rule. The multiplication rule for independent events.
The total probability rule.
Bayes' theorem uses the ______ _______ ______ to update the probability of an event that has been affected by a new piece of evidence.
Total Probability rule
In general, the conditional probability will be greater than the unconditional probability. True or false?
True
Suppose that for a given year there is a 2% chance that your desktop computer will crash and a 6% chance that your laptop computer will crash. Moreover, there is a 0.12% chance that both computers will crash. Is the reliability of the two computers independent of each other?
Yes, because the P(D∩∩L) =.0012 and probability that both computers will crash is also .0012
Anthony feels that he has a 75% chance of getting an A in statistics and a 55% chance of getting an A in Managerial Economics. He also believes that he has a 40% chance of getting an A in both classes. What is the probability that he gets an A in at least one of these courses?
You use the addition rule because it's an OR statement. Another way of saying it is "What is the probability that he gets an A in either Statistics OR Managerial Economics?" Applying the Addition Rule: https://imgur.com/a/uUKwTFr
The addition rule and formula
allows us to find the probability of the union (A OR B, "OR", "At least") of two events. ; P (A ∪ B) = P (A) + P (B) - P(A ∩ B)
A probability based on logical analysis rather than on personal judgment or observation is best referred to as a(n)
classical probability
Marginal probability
the probability of a single event without consideration of any other event
Joint probability
the probability of two events occurring together
Sample Space and Example
the set of all possible outcomes denoted by S. Ex of a sample space representing all possible letter grades in a course S = {A, B, C, D, F}
Events are collectively exhaustive if __________.
they contain all outcomes of an experiment
Total Probability Rule
used to determine the unconditional probability of an event, given conditional probabilities
Which of the following are mutually exclusive events of an experiment in which two coins are tossed? {TT, HH} and {TT} {HT, TH} and {TH} {TT, HT} and {HT} {TT, HH} and {TH}
{TT, HH} and {TH} ; Events are mutually exclusive if they do not share any common outcome of a random experiment.
