15. In class, we have discussed the interpretation of probability as a relative

Question

15. In class, we have discussed the interpretation of probability as a relative frequency ("objective"). Another interpretation of probability is as a measure of belief about the likelihood an event will occur in a particular replication -subjective"). Suppose you ask a friend about the chance of rain tomorrow. Not having had this class, your friend replies with the belief that there is · a 30% chance of rain today, ·a 40% chance of rain tomorrow, ·a 20% chance of rain both today and tomorrow, and ·a 60% chance of rain either today or tomorrow Are these subjective probabilities consistent with the axioms of probability? Why or why not? If not, propose new subjective beliefs that are consistent with the axioms of probability.

Explanation / Answer

Let A be the event that it will rain today and B be the event that it will rain tomorrow

Here, P(A) = 0.3

P(B) = 0.4

Also given that P(A and B) = 0.2

According to the axiom, P(A or B) = P(A) + P(B) - P(A and B)

= 0.3 + 0.4 - 0.2

= 0.5

But its given that, P(A or B) = 0.6

So, this subjective probability is inconsistent with the axioms of probability.

To make it consistent, we shall change the last statement as,

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