Exercise Sheet I
Probability Theory Review
1) Mathematical Basics (2 points)
Use set theory and the definition of probability functions to show that:
P(A ∪ B) = P(A)+ P(B)− P(A ∩ B)
2) Indepedence of Events (8 points)
Consider a fair 6-sided die whose sides are numbered from 1 to 6 and each die roll is independent of the other rolls. In an experiment that consists of rolling the die twice, the following events can be defined
A : The sum of the two outcomes is at least 10
B : At least one of the two rolls resulted in 6 C : At least one of the two rolls resulted in 1
D : The outcome of the 2nd roll was higher than the 1st roll E : The difference between the two roll outcomes is exactly 1
(a) Compute the probabilities P(A), P(C), and P(E).
(b) Is event A independent of event B? (c) Is event A independent of event C?
(d) Are events D and E independent?
3) Bayes Theorem (4 points)
Suppose we are interested in a test to detect a disease which affects one in 100,000 people on average. A lab has developed a test which works but is not perfect. If a person has the disease, it will give a positive result with probability 0.97; if they do not, the test will be positive with probability 0.007. You took the test, and it gave a positive result. What is the probability that you actually have the disease?
4) Random Variables (6 points)
Are X and Y , as defined in the following table, independently distributed? How did you check?
x
0 0 1 1
y
0 1 0 1
p(X = x,Y = y)
0.32 0.08 0.48 0.12
Justify your answers using the laws of probability and the definition of probabilistic independence.
