Question

I can't figure out how to create the decision tree with the sending a messaged situation!

The crew of Endurance can visit two planets (Mann’s and Edmunds’). They can choose to visit neither planets, one of the two planets, or both planets. The characteristics of Mann’s planet are below:

• 30% chance of finding a perfectly habitable planet

• can support all of Earth’s current population if it is

• can support none of Earth’s population if it is not

And the characteristics of Edmunds’ planet are below:

• 50% chance of finding a perfectly habitable planet

• can support 50% of Earth’s current population if it is (because it is not as large as Mann’s planet)

• can support 20% of Earth’s current population if it is not (because it is still partially habitable)

The crew also needs to decide when to send a message to Earth to let them know which planet to migrate to. The possible outcomes for the different time steps of when they send that message are below:

• If they send the message before visiting both planets, none of the Earth’s population would have perished on Earth before receiving that message.

• If they send the message after visiting only one planet (either one), 10% of the Earth’s population would have perished on Earth before receiving that message.

• If they send the message after visiting both planets, 25% of the Earth’s population would have perished on Earth before receiving that message.

What should the crew do to save as many of Earth’s population as possible? Specifically, which planet or planets should they visit, if any and in what order, and when should they send the message to Earth? Draw a decision tree to solve this problem.

Answer 1

Start with No. of planets to be visited => 0, 1 or 2

In case of 0, calculated expected survival

In case of 1, segment the tree further => Which planet to visit? Mann's => Calculate the survival probability, or in case of Edmund's segment the tree further

or Edmund's and calculate the survival probability in both the cases. In the case of Edmi

In case of two, segment the tree further => Where to go first? Mann's or Edmund's & calculate the survival probability in both the cases.

Choose the branch with maximum survival probability

1. How many planets do they visit before sending the message?

i) 0 Planets => 100% Population remains on earth

E(Survival) = (100% Population remains on earth)*[E(Survival at planet 1) + E(Survival at Planet 2)]

=> (100% Population remains on earth)*[30%*100% + (50%*50% + 50%*20%)]

=> 65%

ii) 1 planet => 90% Population remains on earth

a.) Mann's => E(survival) =>

(90% Population Remains)*[(30% Chance of being Habitable => Ask population to migrate to Mann's)*(100% Population can be accomodated) + (70% Chance of being Unhabitable => Ask population to migrate to Edmund's)*(35% Excpected survival (by previous part))]

= 90%*(30%*100% + 70%*35%) = 49.05%

b) Edmund's => Expected Survival

= (90% Population Remains)*[(50% Chance of being Habitable => Ask population to migrate to Edmund's)*(50% Population can be accomodated)

You can calculate th rest similarly! I'm also attaching a hand-drawn diagram