If something has a 5% chance of success, and I’ve failed 19 consecutive times, the next time independently still has a 5% chance of success, fairly low, rather than a guaranteed 100% chance of success in view of the previous attempts. How does this relate to updating percentages in light of new evidence, or is that something separate?
In: 6
So you’re getting a couple things confused here. There are actually 3 different situations you’re sort of conflating:
Situation A: You write the numbers 1-20 on some pieces of cardboard, and put all the pieces in a jar. You mix them up thoroughly and pick a piece without looking; then you look at it, see it’s not a 20, then rip it to shreds and throw it in the trash. Repeat the previous sentence 19 times.
Situation B: You write the numbers 1-20 on some pieces of cardboard, and put all the pieces in a jar. You mix them up thoroughly and pick a piece without looking; then you look at it, see it’s not 20, and put it back in the jar. Repeat the previous sentence 19 times.
Situation C: I pick a secret number N, write the numbers 1-N on some pieces of cardboard, and put all the pieces in a jar. You mix them up thoroughly and pick a piece without looking; then you look at it, see it’s 10 or smaller, and put it back in the jar. Repeat the previous sentence 19 times.
After each situation, you participate in an auction. The prize in the auction is a piece of paper that says, “I will mix the pieces in the jar fairly and draw a piece. If the piece says 20, I will pay you $100. – The Bank”.
Then the probability is how much you should be willing to pay for that piece of paper.
– For Situation A, you should be willing to pay $100. After 19 pieces have been drawn *and shredded* the remaining piece is definitely the 20.
– For Situation B, you should be willing to pay $5. There are 20 pieces equally likely to be chosen, and only one of them has the winning text.
– For Situation C, it’s a little fuzzier and the math is a bit complicated. Basically you assume (before you draw the pieces of paper) how likely I am to pick various values of N, then update those assumptions based on what the pieces of paper say. If you draw 19 pieces of paper and none of them are greater than 10, the Bank’s contract is nearly worthless, as it’s extremely unlikely I wrote N=20; you’d be vastly overpaying for The Bank’s contract even if you only paid $0.01.
Situation A is like a card game. “Drawing without replacement” is jargon for Situation A.
Situation B is like an experiment where the outcome’s determined by a repeatable process with randomized elements. Jargon for Situation B is “Drawing with replacement” or “Bernoulli trial.”
Situation C is like updating your worldview based on new evidence. Jargon for Situation C is “Bayesian inference” or “updating your prior”.
And now we get to the word “Independence.” Independence refers to multiple random events that don’t affect each other. (In other words, knowing the outcome of one event doesn’t change the probability you assign to the other event.)
In B the draws are independent. In A the draws are not independent. In C.. It’s complicated; if you know N it’s Situation B (with N instead of 20) and they’re independent. If you don’t know N, draws aren’t independent, because they’re both influenced by the unknown N (which you treat as another random variable).
The way your OP is worded, it’s not clear whether “independently” is part of the problem statement, or part of the solution.
If you intended to say this:
– Version 1: *Problem:* Trying has a 5% chance of success, and I’ve tried and failed 19 consecutive times. What’s my chance of failing on the next try? *Solution:* Independently, there’s still a 5% chance of success.
– Version 2: *Problem:* Trying has a 5% chance of success, and I’ve tried and failed 19 consecutive times. Tries are independent. What’s my chance of failing the next time? *Solution:* There’s still a 5% chance of success.
In Version 2 the solution’s correct and all is well.
In Version 1, things are.. problematic. You don’t know whether you’re in Situation A or Situation B; the problem didn’t tell you. If you only use the information given in the problem statement, you have to say “I don’t know because the problem didn’t include enough information to tell me whether I’m in Situation A or Situation B.”
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