Question

Come up with an example of two mutually exclusive and collectively exhaustive events of your choice and assign their probabilities. Come up with a conditioning event and demonstrate how conditioning can change the probability of these events. Explain why that is the case. 2.

Answer 1

Let us define an event where a doctor will diagnose a disease correctly or not. The event that he correctly diagnoses a patient and the event that he fails to correctly diagnose are two mutually exclusive and collectively exhaustive events, since when one event occurs, the other doesn't and both events make up the sample space we are interested in.

A: Disease is diagnosed correctly by the doctor, with a probability of 0.6.

B: Disease is diagnosed incorrectly by the doctor

lets also define a third event, C : A patient of the doctor who had the disease dies.

P(A) = 0.6, therefore, P(B) = 1 - 0.6= 0.4

Suppose the chances that a patient will die by his treatment after correct diagnosis is 40% and that of by wrong diagnosis is 70%, which means P(C| A)= 0.4 and P(C|B)= 0.7.

P(C) = P(C|A)*P(A) + P(C|B)*P(B) = 0.6*0.4 + 0.4*0.7 = 0.52

By Bayes theorem,

P(A|C) = (0.6*0.4)/0.52 = 0,.46

so, from the above example we can see that conditioning event C changes the probability of event A, earlier P(A)= 0.6 now after conditioning it is P(A|C) = 0.46. This is because when we use conditional probability, the number of events conforming to the condition is reduced and we find the probability of an outcome based on partial information. Conditioning on event C means changing the sample space to event C.