4 ms·

I had a hard time understanding part of the explanation to Problem 6 [1]. The statement: > Consider the following game. You flip a coin until you get a tails.

I had a hard time understanding part of the explanation to Problem 6 [1]. The statement: > Consider the following game. You flip a coin until you get a tails. The number of dollars you win equals the number of coins you end up flipping. (So if you immediately get a tails, you win one dollar; if you get one heads before a tails, you win two dollars, etc.) What is the expectation value of your winnings? And then the beginning of the explanation [2]: > There is a 1/2 chance that you win one dollar, a 1/4 chance that you win two dollars, a 1/8 chance that you win three dollars, etc. Therefore... It took me a few times of reading it to understand this. If you choose to play this game, you have a 100% chance of making at least $1. Why state it that there's a 1/2 chance you win $1? It doesn't seem to help answer the problem because assuming you're playing you will always always win $1. The reason they say this (I think) is that there's really a 50% chance you win $1 and a 50% chance you win $2. But again, this doesn't help you with expected winnings - you know you've got something on the first pass. Maybe it help explains Part 2? Anyway, cool concept! [1]: http://web.physics.harvard.edu/uploads/files/undergrad/probweek/prob6.pdf http://web.physics.harvard.edu/uploads/files/undergrad/probw... [2]: https://www.physics.harvard.edu/uploads/files/undergrad/probweek/sol6.pdf https://www.physics.harvard.edu/uploads/files/undergrad/prob...

mlevental 8y ago
what's the sum of n/2^n? Edit: generatingfunctionology to the rescue
jackcarter 8y ago
> Why state it that there's a 1/2 chance you win $1? It doesn't seem to help answer the problem because assuming you're playing you will always always win $1. The reason they say this (I think) is that there's really a 50% chance you win $1 and a 50% chance you win $2. There is a 50% chance of winning exactly $1. There's a 25% chance of winning exactly $2. It sounds like you might be interpreting it to mean "at least $1" instead of "exactly $1".
- tr33house 8y ago
  Understanding this really helped me understand probability in college. The __exactly__ is the difference
- pseudoramble 8y ago
  Thanks - This is exactly what I was doing thinking back on it! I found it surprising because I landed on the same answer they did even with my misunderstanding of the statement. Ultimately I ended up figuring out this sequence: > 1 + 1/2 + 1/4 + 1/8 + ... = 2 But this now seems like I got kind of lucky, and it might not always hold up generally.

mlevental 8y ago

what's the sum of n/2^n? Edit: generatingfunctionology to the rescue

jackcarter 8y ago

> Why state it that there's a 1/2 chance you win $1? It doesn't seem to help answer the problem because assuming you're playing you will always always win $1. The reason they say this (I think) is that there's really a 50% chance you win $1 and a 50% chance you win $2. There is a 50% chance of winning exactly $1. There's a 25% chance of winning exactly $2. It sounds like you might be interpreting it to mean "at least $1" instead of "exactly $1".

tr33house 8y ago
Understanding this really helped me understand probability in college. The __exactly__ is the difference
pseudoramble 8y ago
Thanks - This is exactly what I was doing thinking back on it! I found it surprising because I landed on the same answer they did even with my misunderstanding of the statement. Ultimately I ended up figuring out this sequence: > 1 + 1/2 + 1/4 + 1/8 + ... = 2 But this now seems like I got kind of lucky, and it might not always hold up generally.