dial911book
New member
- Joined
- Jun 17, 2019
- Messages
- 4
Thank you for helping with this, I have an educated guess how to do it but would really appreciate an expert's input. Start with a 52 card standard deck. I write on a piece of paper one long sequence that accounts for all 52 cards. To visualize, I write 2C, 4H, 6D, 8S, AC, KH, 9H, … all the way through. So I know what the target sequence is.
Now -- another person starts dealing out one card at a time from a shuffled 52 card deck. The first card is, say, 5C. Oops - 5C is not the correct first card in my predefined sequence. So she puts the card back in the deck, shuffles, and deals out a 2C. Fine, that matches the predefined sequence. She deals at random the next card. Oops, it is a 10S. So she takes both of the cards back into the deck, shuffles, and starts over.
Question: How many of these deals, where there is no guarantee that the first or second or third … or 51st card, will be the correct one in the sequence -- and whenever the wrong card comes into the sequence she is dealing, she takes all of the cards back, shuffles, and starts again -- would be expected to occur (on average) to obtain all 52 card dealt out in the specified order without a reshuffling?
(If you can show how you made the calculation, that will greatly help me, so I can generalize to other related sorts of problems.)
(If it is simpler to show how it is done for, say, 5 distinct cards, that's fine, if I can generalize the formula )
(It occurs to me that the solution is a number much larger than 52!, which is the number of discrete arrangements of all 52 cards. I just don't know how to think about the problem of restarting from scratch when the sequence is part way but then fails.)
-Richard
Now -- another person starts dealing out one card at a time from a shuffled 52 card deck. The first card is, say, 5C. Oops - 5C is not the correct first card in my predefined sequence. So she puts the card back in the deck, shuffles, and deals out a 2C. Fine, that matches the predefined sequence. She deals at random the next card. Oops, it is a 10S. So she takes both of the cards back into the deck, shuffles, and starts over.
Question: How many of these deals, where there is no guarantee that the first or second or third … or 51st card, will be the correct one in the sequence -- and whenever the wrong card comes into the sequence she is dealing, she takes all of the cards back, shuffles, and starts again -- would be expected to occur (on average) to obtain all 52 card dealt out in the specified order without a reshuffling?
(If you can show how you made the calculation, that will greatly help me, so I can generalize to other related sorts of problems.)
(If it is simpler to show how it is done for, say, 5 distinct cards, that's fine, if I can generalize the formula )
(It occurs to me that the solution is a number much larger than 52!, which is the number of discrete arrangements of all 52 cards. I just don't know how to think about the problem of restarting from scratch when the sequence is part way but then fails.)
-Richard