(1) Example. Consider a series of hands dealt at bridge. Let A be the event that in a given deal each player has one ace. Show that the probability that A occurs at least once in seven deals is approximately 1/2.
This is a worked example from Probability and random processes, by Grimmet. As part of my self study I am reviewing probability material I learned early on in my undergrad and grad career. I think I understand the mechanics of the problem, but I think my confusion arises from some misunderstanding of what is meant by a hand vs a deal.
I can compute the number of ways of giving 13 cards to any 1 of the 4 players (this is what I interpreted as hand), this number I agree with authors is $ \binom{52}{13} = \frac{52!}{ 13!(52-13)!}$.
What I interpret as deal, was the act of drawing an individual card handing it it player A, then another and handing it to player B, then another and handing it to player C, and the last to player D. I would consider this to be 4 deals. You repeat this cycle of 4 deals 13 more times till all four players have 13 cards. This seems incorrect.
But it seems the authors consider a deal to be the final result, i.e, each player having 13 cards, (or at least how I understand it). I would like to make sure I am understanding this correctly.
Doing $P(A : \text{each player has one Ace in given deal}) = \frac{4! \frac{48!}{(12!)^4}}{\frac{52!}{(13)!^4}} \approx \frac{1}{10}$
The $P(A^c) = \frac{9}{10}$ $7$-deals then means you do this process 7 times, i.e deal out 13 cards each to each of the 4 players, then take them back re-shuffle and repeat 7 times. Then $P (\text{A occurs at least once}) = 1- \big(\frac{9}{10}\big)^7 \approx \frac{1}{2}$
Any insights would be greatly appreciated.
A "deal" usually refers to the complete distribution of cards, and can be synonymous with "round" or "hand". Distributing all 52 cards to 4 players does require dealing 52 cards, but it is usually described as a single deal, not 4 deals or 13 or 52. Each card dealt is not itself "a deal".
When playing cards, you may hear someone complain about a "bad deal" - they are complaining about their hand as a whole, not a single particular card. If someone wants "one more deal", they want to play another round, not get one more card.