Tyranthraxus posted...Sativa_Rose posted...
RedZaraki posted...
An explicit explanation.
Let's draw numbers on the balls to track them, shall we?
Box 1, Gold 1 = #1
Box 1, Gold 2 = #2
Box 2, Gold = #3
Box 2, Silver = #4
Box 3, Silver 1 = #5
Box 3, Silver 2 = #6
You choose a box at random. This is 1/3 probability. You choose a ball at random. This is 1/2 probability (within that box).
Thus, you have a 1/6 probability of pulling any specific ball here.
HOWEVER. The rules state YOU PULLED A GOLD BALL. Yes? Guaranteed, you are holding either ball #1, #2, or #3 in your hand. AND they all had equal probability of occurring. Thus, 1 out of 3 odds each.
So the odds you are holding each ball right now is:
#1 = 1/3
#2 = 1/3
#3 = 1/3
#4 = 0 (we did not pull a silver ball)
#5 = 0 (we did not pull a silver ball)
#6 = 0 (we did not pull a silver ball)
Everyone follow so far?
Let's look at what happens in each of the three cases. GIVEN that you know you pulled a gold ball:
1/3 times you Pulled #1: Your next pull is guaranteed to be #2, also a gold ball. = 100% gold
1/3 times you Pulled #2: Your next pull is guaranteed to be #1, also a gold ball. = 100% gold
1/3 times you Pulled #3: Your next pull is guaranteed to be #4, a silver ball. = 0% gold
Let's add these up.
(1/3 * 100%) + (1/3 * 100%) + (1/3 * 0%) = (1/3 * 1.0) + (1/3 * 1.0) + (1/3 * 0.0) = 2/3
Okay I believe this is correct. Another way to think of the final answer is to say that if one were to run this experiment thousands of times, they would pull out another gold ball 2/3 of the time that the first one they pull out is gold.
Probability gets wonky when you add conditions.
Like, for example, if I were to say how many heads would you get on average in a single coin flip. The answer is 1/2 except you can't flip a half of a head so it's not a real result, but that's how the math works out.
People tend to think about these problems as tangible things with a sequence of events within a specific occurrence and not as a more general abstract math problem.
You could repeat this exercise with 1 million extra boxes of 2 silver balls and the result is unchanged.
Exactly.
Here's an example:
You flip a coin. The first result is Heads. What is the probability that the second flip is also heads?
The answer is still 1/2. Every single flip is 1/2.
It doesn't matter that your previous flip was heads. Future flips do not tend to be more likely tails because you got heads already. They are all equally heads and tails forever.
All probability means, is that given an INFINITE dataset, the result set TRENDS towards 1/2 heads and 1/2 tails.
---
https://imgur.com/jKWMT8a