Probabilities Flashcards
basic probability rules (combining probabilities), probaility mass functions, odds ratio
What values can probabilities take?
Between 0 and 1
- Pr = 0 is impossible, will never happen
- Pr = 1 will definitely happen
What is the complement of an event F?
Pr(F bar) = 1 - F
What is the intersection of F and G?
- probability F and G both occur
- Pr(F∩G)
How is the intersection of F and G calculated when the two values are independant?
Pr(F∩G) = Pr(F) x Pr(G)
What is the union of F and G?
- either F or G occurs
Pr(F∪G) = Pr(F) + Pr(G) - Pr(F∩G)
What happens when F and G are mutually exclusive?
- F and G cannot occur together
- Pr(F∩G) = 0
What is conditional probability?
- Pr of F occuring given G
Pr(F | G) = Pr(F∩G) /Pr(G)
What is the equation for the Probability Mass Function?
Σ pr(x) = P(X=k) + P(X=k+1) + … + P(X=n) = 1
What is the expected value of X and the variance (notation)?
E(X) = Σ x pr(x) var(X) = E[x^2] - (E[x])^2
In the binomial distribution, what does a low probability of success mean? What happens when its 0.5?
p is low: expect right skewed
p = 0.5: expect symmetry
In the binomial distribution, what do p, x and n mean?
p - pr of success
x - x successes in a row
n - number of trials
What are the conditions for the Binomial Distribution?
- two outcomes (eg yes/no)
- fixed number of trials
- independant observations
- constant pr of success
How are the confidence intervals for a binomial distribution calculated?
“CI for a proportion” on formula list.
What do we need to be able to assume that estimates for a proportion are normally distributed?
Require large samples.
When can we not assume that proportions are normally distributed? What do we use to calculate confidence intervals in this case?
- when p is close to 0 or 1 (p < 0.05, p > 0.5)
- Wilson intervals are used instead
How many cases are there for proportions (for calculating the standard error)?
3
- proportions from independant samples
- one sample of size n, several response categories
- one sample of size n, many yes/no items
What is an odds ratio?
- used to quanitfy the extent of the association between two groups
- a relative measure of effect (eg. comparison of an intervention group of a study relative to a control/placebo group)
How is an odds ratio calculated?
- numerator is the odds in group 1
- denominator is the odds in group 2 (/control)
What does an odds ratio of 1 mean?
- outcome is the same in both groups
- no differece between the two groups in the study
What does an odds ratio > 1 mean? What about OR < 1?
OR > 1: intervention is better than control
OR < 1: control is better than intervention
never negative
What is an alternative way to calculate the odds of success (way not on formula list)?
p/(1 - p)
What does an odds of success ratio of 4 mean?
Success is 4x more likely.
How is a probability calculated from its odds?
p = odds/(odds + 1)
How is the standard error for the odds ratio calculated?
se(OR) = sqrt((1/n11) + (1/n12) + (1/n21) + (1/n22))
where n11…n22 are based on the number of observations in each group/outcome category
How is the confidence interval for the odds ratio calculated?
ln(θ) ± z x se(OR)