Hi all. The below is just a question that I've been randomly pondering myself, so I don't really know if there's an official "type" of probability this falls under, so my subject name might be misleading.
The question I have focuses on multiple instances of something, each with its own probability of success. While I was thinking of this as a purely hypothetical, I'm making an example up for this so it makes more sense.
Say you have various machines that manufacture an item, like a plate. Each machine has an independent chance of success/failure in manufacturing a plate successfully. And (for the purpose of this question), if you were to combine machines, the chances increase linearly (so if machine A has a 20% chance to succeed, and machine B has a 30% chance, using both machines together to make a plate will give a 50% chance).
The goal is to create as many plates as possible. Is it thus better to run the machines independently, or to combine them.
Now, the question thus far is rather vague/lose, so I ask myself if "it depends." For example, say each machine only has a 1% chance of success, would the answer to the question be different if each machine had a 20% chance to succeed. So as to not completely confuse myself though, I decided to look at one arbitrary specific scenario.
Machine A: 40% chance to succeed; Machine B: 40% chance to succeed
Remember, the goal is to create as many 'plates' as possible.
Option 1: Combine the machines, so you have a 80% chance to create 1 plate.
Option 2: Run them independently. Now, in my mind, I can't just do 2*40%. Instead, it's more like: (2 C 1) * (0.4)^1 + (0.6)^1 = 1 * 0.4 * 0.6 = 0.24. But that means I only have a 24% chance to get EXACTLY ONE plate. But I can get 2 plates too, which would be: (2 C 2) * (0.4)^2 * (0.6)^0 = 1 * 0.16 * 1 = 0.16. So I have a 16% chance to get TWO plates.
Off the bat, I think to myself, "24% to get 1, 16% to get 2.. that's no very good chances there. Combined, I have a 40% chance to get AT LEAST ONE plate". But this doesn't help me answer the question of which option to get.
So I think to myself... expected values.
Option 1: 80% * 1 = 0.8
Option 2: 24% * 1 + 16% * 2 = 0.56
That would mean Option 1 is better.
Anyway... using THAT specific scenario, is my logic sound? Or am I thinking about this incorrectly?
The question I have focuses on multiple instances of something, each with its own probability of success. While I was thinking of this as a purely hypothetical, I'm making an example up for this so it makes more sense.
Say you have various machines that manufacture an item, like a plate. Each machine has an independent chance of success/failure in manufacturing a plate successfully. And (for the purpose of this question), if you were to combine machines, the chances increase linearly (so if machine A has a 20% chance to succeed, and machine B has a 30% chance, using both machines together to make a plate will give a 50% chance).
The goal is to create as many plates as possible. Is it thus better to run the machines independently, or to combine them.
Now, the question thus far is rather vague/lose, so I ask myself if "it depends." For example, say each machine only has a 1% chance of success, would the answer to the question be different if each machine had a 20% chance to succeed. So as to not completely confuse myself though, I decided to look at one arbitrary specific scenario.
Machine A: 40% chance to succeed; Machine B: 40% chance to succeed
Remember, the goal is to create as many 'plates' as possible.
Option 1: Combine the machines, so you have a 80% chance to create 1 plate.
Option 2: Run them independently. Now, in my mind, I can't just do 2*40%. Instead, it's more like: (2 C 1) * (0.4)^1 + (0.6)^1 = 1 * 0.4 * 0.6 = 0.24. But that means I only have a 24% chance to get EXACTLY ONE plate. But I can get 2 plates too, which would be: (2 C 2) * (0.4)^2 * (0.6)^0 = 1 * 0.16 * 1 = 0.16. So I have a 16% chance to get TWO plates.
Off the bat, I think to myself, "24% to get 1, 16% to get 2.. that's no very good chances there. Combined, I have a 40% chance to get AT LEAST ONE plate". But this doesn't help me answer the question of which option to get.
So I think to myself... expected values.
Option 1: 80% * 1 = 0.8
Option 2: 24% * 1 + 16% * 2 = 0.56
That would mean Option 1 is better.
Anyway... using THAT specific scenario, is my logic sound? Or am I thinking about this incorrectly?