Cardplayer
New member
- Joined
- Jun 18, 2015
- Messages
- 1
When I play the game Hearts with my parents, we are each dealt 17 cards with one remaining card put into the "cat". The person who takes the first trick gets to look at this card, but nobody else. The specific rules to the game aren't important, other than that each player has to follow whatever suit is led.
Now before any of us look at our cards, there is clearly a 1 in 4 chance of the card being any particular suit. But once we've each looked at our cards and begun to play the odds change. Two of the players usually can't be certain until the very end of the hand what that specific card is. My problem is that I've really confused myself about how to these odds change as the hand plays out.
Lets say you're dealt 4 clubs, 4 hearts, 4 spades, and 5 diamonds. At this point there are 35 unknown cards, so the probability of it being a club, for instance, are 9/35 = 0.257. Now you start playing the game and lead a club (you don't get to look in the "cat"). The other players follow suit and these 3 clubs are removed from play. There are 33 unknown cards now, but only 7 of which are clubs, so the probability has dropped down to 7/33 = 0.212. Someone else lead clubs two more times and its dropped to 3/29=0.103. If you lead your last club and they both follow suit it appears there is only a 1/27=0.037 chance of a club being in the "cat", but it still feels like it should be closer to 1 in 4.
My reasoning is as follows; if I randomly deal one card face down on a table from a full deck and then ask you to remove twelve clubs from the deck, I am extremely confident that one out of four times the card on the table will be a club. Removing those other cards afterwards did nothing to affect the odds. I could have you remove all 51 cards, and it still wouldn't affect it.
As I write this I can see that in the example where I got 0.037 I've actually calculated the probability of any specific card in my opponents hand or the cat being the missing club, but I just can't seem to wrap my head around what I've done wrong here. Leading clubs four times or leading diamonds four times and having the opponents follow suit isn't that rare so it just can't be the fact that it reduces the odds of a card being in the deck so drastically. I feel like it will be a DOH! moment when someone points it out. Thanks.
Now before any of us look at our cards, there is clearly a 1 in 4 chance of the card being any particular suit. But once we've each looked at our cards and begun to play the odds change. Two of the players usually can't be certain until the very end of the hand what that specific card is. My problem is that I've really confused myself about how to these odds change as the hand plays out.
Lets say you're dealt 4 clubs, 4 hearts, 4 spades, and 5 diamonds. At this point there are 35 unknown cards, so the probability of it being a club, for instance, are 9/35 = 0.257. Now you start playing the game and lead a club (you don't get to look in the "cat"). The other players follow suit and these 3 clubs are removed from play. There are 33 unknown cards now, but only 7 of which are clubs, so the probability has dropped down to 7/33 = 0.212. Someone else lead clubs two more times and its dropped to 3/29=0.103. If you lead your last club and they both follow suit it appears there is only a 1/27=0.037 chance of a club being in the "cat", but it still feels like it should be closer to 1 in 4.
My reasoning is as follows; if I randomly deal one card face down on a table from a full deck and then ask you to remove twelve clubs from the deck, I am extremely confident that one out of four times the card on the table will be a club. Removing those other cards afterwards did nothing to affect the odds. I could have you remove all 51 cards, and it still wouldn't affect it.
As I write this I can see that in the example where I got 0.037 I've actually calculated the probability of any specific card in my opponents hand or the cat being the missing club, but I just can't seem to wrap my head around what I've done wrong here. Leading clubs four times or leading diamonds four times and having the opponents follow suit isn't that rare so it just can't be the fact that it reduces the odds of a card being in the deck so drastically. I feel like it will be a DOH! moment when someone points it out. Thanks.
