Does each wallet have only one private key?

[outatime1]

When a wallet is created, there is a pubic key and private key.  But there is also a character string or phrase, right?  If you have any 2 of the 3 you can access the account, correct?  What I'm trying to figure out is can it be possible for someone else to generate the same public key with a different private key?  And how likely is it for the bitcoin address generators to come up with the same address a second time for a different person?

[achow101]

A wallet is a collection of different private keys. Public keys are derived from private keys. There are 2^256 - 1 possible private keys and thus that many public keys.

Some wallets are Hierarchical Deterministic which means that they have one master private key which other private keys are determined from.

It is possible to generate the private keys to someone else's addresses because technically they are all known, every single number from 0 to 2^256 - 1. However that range is so large that the likelihood of generating the private key to an address in use is very low unless the random number generators used is broken in some way like blockchain.info has done.

[pooya87]

When a wallet is created, there is a pubic key and private key.  But there is also a character string or phrase, right?

AFAIK, all the wallets will create a public key and private key but there are only a few that make a string called seed, that you can use to recover the whole wallet.
you can read about how seed works in Electrum (a wallet that is using seed to generate and restore the wallet) documents: http://docs.electrum.org/en/latest/faq.html#what-is-the-seed

If you have any 2 of the 3 you can access the account, correct?  

the bitcoin in an address can be accessed (spent) having the private key of that address.
and as i said that string (Seed) can be used to recover the wallet and all the private keys in it.

What I'm trying to figure out is can it be possible for someone else to generate the same public key with a different private key?  

no, each private key will create a unique bitcoin address

And how likely is it for the bitcoin address generators to come up with the same address a second time for a different person?

short answer is no.
there is a math that you might find if you search that proves the chance of it happening is super small. in other words you have better chance of getting hit by a meteor than creating a duplicate bitcoin address

[Nandiwal]

One answer of all doubt about hacking a BTC key is "NO". Today and may be after 100 years there is not a single chance.

[Pursuer]

sometimes using google, you can get detailed answers to any question you might have. since the rest of your questions are covered I just wanted to add this here about the possibility of duplicate bitcoin address:
http://bitcoin.stackexchange.com/a/7732
https://bitcointalk.org/index.php?topic=64413.0

[quote https://en.bitcoin.it/wiki/Technical_background_of_version_1_Bitcoin_addresses]
Since Bitcoin addresses are basically random numbers, it is possible, although extremely unlikely, for two people to independently generate the same address. This is called a collision. If this happens, then both the original owner of the address and the colliding owner could spend money sent to that address. It would not be possible for the colliding person to spend the original owner's entire wallet (or vice versa). If you were to intentionally try to make a collision, it would currently take 2^107 times longer to generate a colliding Bitcoin address than to generate a block. As long as the signing and hashing algorithms remain cryptographically strong, it will likely always be more profitable to collect generations and transaction fees than to try to create collisions.
It is more likely that the Earth is destroyed in the next 5 seconds, than that a collision occur in the next millenium.
[/quote]

[outatime1]

Thank you all for the help on this one.  That explains it a lot.  I still need to read up on some of those point to fully understand.

[unholycactus]

Thank you all for the help on this one.  That explains it a lot.  I still need to read up on some of those point to fully understand.

Other people described pretty advanced wallet types. But I'll illustrate the simplest case to you so you can start understanding better.

Wallets are basically just a collection of addresses.
A good wallet uses addresses only once.
Each addresses has a private key.

But it is also possible to generate a bunch of private keys off of one master key.

[Avirunes]

When a wallet is created, there is a pubic key and private key.  But there is also a character string or phrase, right?  If you have any 2 of the 3 you can access the account, correct?  What I'm trying to figure out is can it be possible for someone else to generate the same public key with a different private key?  And how likely is it for the bitcoin address generators to come up with the same address a second time for a different person?

Wallet has many addresses ,its just like a normal wallet having many pockets in which you keep money.First no. Of private key depends on no. of addresses you have like if you have 3 addresses in wallet then you have 3 private keys.Well public key is only shown for showing publicly so no one can access the account from public key.