Probability (Part 2)

• Solutions for Homeworks #1 and 2 are on Blackboard!!!

Definition: The probability of an event not occurring is one minus the probability that it occurs. \[ \mathrm{Pr[{\it not}\ A]} = 1-\mbox{Pr[A]} \]

Definition: The law of total probability is given by \[ \begin{align*} \mathrm{Pr[A]} & = \sum_{B\ \mathrm{in} \ \mathcal{M}}\mathrm{Pr[A \ and \ B]} \\ & = \sum_{B\ \mathrm{in} \ \mathcal{M}} \mathrm{Pr[B]}\ \mathrm{Pr[A\ | \ B]}, \end{align*} \] where \( \mathcal{M} \) is a set of mutually exclusive events such that \[ \sum_{B\ \mathrm{in} \ \mathcal{M}}\mathrm{Pr[B]} = 1 \]

Definition: A probability distribution is a list of the probabilities of all mutually exclusive outcomes of a random trial.

Compare to:

Definition: A probability distribution (or relative frequency distribution) is a list of the probabilities of all values of a random variable in a sample or population.

Probability densities

1. Write out the probability that you're are being asked to find. Is it a conditional probability? AND? OR?
2. Identify probabilities that you are given (again, are these conditionals? ANDs? ORs?)
3. Draw a probability tree (if appropriate)

Smoking and cancer contingency table

            health
status       cancer not cancer    Sum
  smoker       8944      43056  52000
  not smoker    624      47376  48000
  Sum          9568      90432 100000

Question: What is Pr[smoker]?

Answer: 52000/100000 = 0.52

Question: What is Pr[cancer]?

Answer: 9568/100000 = 0.09568

Smoking and cancer contingency table

            health
status       cancer not cancer    Sum
  smoker       8944      43056  52000
  not smoker    624      47376  48000
  Sum          9568      90432 100000

Question: What is Pr[cancer | smoker]?

Answer: 8944/52000 = 0.172

Question: What is Pr[smoker | cancer]?

Answer: 8944/9568 = 0.9347826

Smoking and cancer contingency table

            health
status       cancer not cancer    Sum
  smoker       8944      43056  52000
  not smoker    624      47376  48000
  Sum          9568      90432 100000

Question: What is Pr[smoker AND cancer]?

Answer: 8944/100000 = 0.08944