Question

It’s the End of the World As We Know It. The potential for an apocalypse worries me. What worries me most is that it could include any number of disasters. There is a 10% chance of a zombie infection. There is a 12% chance of Ragnorak (that’s the Norse apocalypse). There is an 8% chance of a UFO invasion6 . Heck, it could be like some show on the SyFy channel and all 3 might happen at the same time. (Translation: All three events are independent and not mutually exclusive.) (a) What is the probability that we will have an apocalypse of one sort or another? (b) I’m also a part-time actuary and in my need to prepare, I’ve decided to only prepare for those events which have a 0.9% chance or greater of occurring. Which types of apocalypses and combinations of apocalypses do I need to prepare for?

Answer 1

a) 0.2713

b)Zombie Infection and Ragnarok, Ragnarok and UFO invasion.

Explanation:

We are given the following information in the question:

There is a 10% chance of a zombie infection. There is a 12% chance of Ragnarok . There is an 8% chance of a UFO invasion.

Thus, we can write:

P(A) = P(Zombie Infection) = 10% = 0.10

P(B) = P(Ragnarok) = 12% = 0.12

P(C) = P(UFO Invasion) = 8% = 0.08

It is given that all three events are independent.

a) P(an apocalypse of one sort or another)

`P(A \cup B \cup C) = P(A) + P(B) + P(C) + -P(A \cup B) - P{A \cup C) - P(B \cup C)`

Since the events are independent

`P(A \cup B \cup C) \\\\=P(A) + P(B) + P(C) - (P(A)* P(B) + P(A)* P(C) + P(B)* P(C))\\\\=0.10 + 0.12 + 0.08 - (0.10* 0.12) - (0.10* 0.08) - (0.08* 0.12)\\\\= 0.2713`

b)

`P(A \cap B) = P(A)* P(B) = 0.10* 0.12 = 0.012\\P(A \cap C) = P(A)* P(C) = 0.10* 0.08 = 0.008\\P(B \cap C) =P(B)* P(C) = 0.12* 0.08 = 0.0096`

Now, want a combination of apocalypse such that probability is greater than or equal to 0.009.

P( Combination ) > 0.009

Thus, the possible combination of apocalypse are:

Zombie Infection and Ragnarok

Ragnarok and UFO invasion.