Week 3: Probability Flashcards
What is the basic definition of probability?
The measure of likeliness that an event will occur or a certain condition will be met
What are the 3 categories of probability?
Frequency based probability
Model based probability
Subjective probability
What is frequency-based probability based on? Provide an example
It is based on a number of occurrences over a long sequence of trials, and they have large amounts of data
Provide an example of frequency based probability
51% of babies born are female = probability of female birth is 51%
A treatment has shown a 37% success rate in treating a condition
What is model based probability based on?
Probabilities are based on known mathematical models
Provide an example of model-based probability
Parent A is AB, Parent B is O = 50% chance each child will be type A and 50% type B
Two non-cystic fibrosis parents have child with CF, then there is 25% chance subsequent children will have it
What is subjective probability based on and what makes it unique?
It is based on beliefs and observations of a person (or group).
Unique: Can differ depending on who is making the probability statement, it changes with new information.
Provide an example of subjective probability
Based on initial exam, I believe there is a 70% chance the injury will now require surgery
Now that I’ve seen the x-ray, there is a 90% chance you need surgery
What numbers is P(A) always between? What does denote an event mean?
P(A) is always between 0 and 1 (0% and 100%)
Denoting an event is written as A’ in a formula (Aprime) and it means that the event does not occur, or whatever is the opposite of the primary event
What are the equations that sum up the following:
- P(A), P(B), P(A and B), P(A or B)
- A and A’
P(A or B) = P(A) + P(B) - P(A and B)
P(A and B) = P(A) + P(B) - P(A or B)
P(A or A’) = 1
P(A and A”) = 0
What is conditional probability? What is the notation?
Is the probability of one event occurring given that another even has (or will) occur
Notation: P(A|B)
What does it mean if the event is independent?
It implies that one event “has nothing to do” with the other. One event’s probability is unaffected by the occurrence (or non-occurrence) of the other
What does a mutually exclusive event mean?
Two events are mutually exclusive if they both cannot occur
What is Bayes Theorem used for?
To compute probabilities in diagnostics. Useful to know the probability of disease given a positive test result
What is the longer (more useful) formula for Bayes Theorem?
P(A|B) = P(B|A)P(A) / P(B|A)P(A)+P(B|A’)*P(A’)
Solve the Following, Show your work:
A drug has two potential side effects, nausea and insomnia. 20% of patients gets nausea, and 60% of patients get insomnia, while 10% get both. What is the probability that a patient taking this drug will get at least one of these side effects?
P(N or I) = P(N) + P(I) - P(N and I)
= 0.2 + 0.6 - 0.1
= 0.7 or 70%
Is the following statement independent or mutually exclusive?
A doctor gets x-rays on 70% of his patients, regardless of whether they are male or female.
This is an independent statement, as the event that an x-ray will be ordered is unaffected by if the patient is male or female
Solve the following:
A doctor requests x-rays for 70% of patients, regardless of gender. 50% are female. Let F be the event that the next patient is male, and X be the event the doctor requests an x-ray on the next patient.
Equations: P(A|B)=P(A); P(B|A)=P(B), P(AandB)=P(A)P(B), P(AorB)=P(A)+P(B)-P(A)P(B)
P(F)= P(F|X) = 50%; P(X) = P(X|F) = 70%
P(F and X) = P(F)P(X) = 0.50.7 = 0.35 or 35%
P(F or X) = 0.5 + 0.7 - 0.5*0.7 = 0.85 or 85%
Is the following statement independent or mutually exclusive?
The events that a mother is blood type AB and her son is blood type O
Mutually exclusive because an AB blood type mother cannot have a type O child
Another example is a doctor saying to a patient that if they don’t operate the patient will never walk again
Solve the following:
A doctor does a prostate exam on 40% of his patients, but never when they are female. 50% of his patients are female. Let F be the event that the next patient is female, and E be the event that a prostate exam will be done on the patient.
Equations: P(A|B), P(B|A), P(AandB), P(AorB)
P(F)=50% P(E)=40% P(F|E)=0 P(E|F)=0
P(FandE) = 0
P(ForE) = 0.5+0.4 = 0.9 or 90%
Solve the following question using Bayes Theorem:
The prevalence of a disease in a population is 10 in 100, and a specific test has a 70% chance of being correct (positive or negative), what is the probability a patient has the disease after testing positive?
Formula: P(A|B) = P(B|A)P(A) / P(B|A)P(A)+P(B|A’)P(A’)
= 0.70.1/0.70.1+0.30.9
= 0.07/0.34
=0.206 = 20.6%
