Question

For a roll with two dice, the following events are considered:
A: The sum of eyes is greater than 7.
B: Exactly one of the two numbers is a 5.
C: No 1 is rolled.
a) Calculate the probabilities
P (A), P (B), P (C), P (A ∩ B), P (A ∩ C), P (B ∩ C), P (A ∪ B), P (A | B), P (A | C), P (C | A), P (B | C).
b) Are events A and B independent or disjoint?

In a hall, there are four machines working independently of each other, which do not fail within a certain time span with the probabilities 0.9, 0.95, 0.8 and 0.85, respectively.
Calculate the probability that during this period
a) all four machines work b) no machine works
c) exactly one machine works d) exactly two machines work
e) exactly three machines work f) at least one machine works!

A device consists of 100 independent modules of equal functionality. Zk be that
Event that the kth group works reliably.
a) What is the probability that the device will work reliably at P (Zk) = 99%?

b) How big must P (Zk) be, so that P (ZGeraet) = 90%?

From a cancer test are given:
Events: T: test result positive, i. Suspected cancer
K: test subject krebskrank
Probability values: P (T | K) = P (Tc | Kc) = 0.95, P (K) = 1/200
Calculate P (T) and P (K | T) and interpret the results!

Answer 1

Q1)

a) When two dies are rolled there are 36 outcomes possible,

For the set A the outcomes are (2,6), (6,2), (3,5), (5,3), (4,4), (3,6), (6,3), (4,5), (5,4), (4,6), (6,4), (5,5), (5,6), (6,5), (6,6)

So,

For the event B the possible outcomes are (1,5), (2,5), (3,5), (4,5), (6,5), (5,1), (5,2), (5,3), (5,4), (5,6)  

So

For the event C the possible outcomes are (2,2), (2,3), (2,4), (2,5), (2,6), (3,2), (3,3), (3,4), (3,5), (3,6), (4,2), (4,3), (4,4), (4,5), (4,6), (5,2), (5,3), (5,4), (5,5), (5,6), (6,2), (6,3), (6,4), (6,5), (6,6)

So,

b) No, events A and B are not independent nor disjoint,

as and