Round and Round They Go...

If they should cut, nobody knows....or do we?

Two people are playing a card game with 18 Black cards, and 3 red cards. The red cards are randomly mixed into the black cards as a group ( the red cards are all together somewhere in the deck at the start of the game example of possible configuration: BBBBBRRRBBBBBBBBBB )

The game is played by drawing cards ( without replacement) from the top of the deck one by one until a red card is drawn. The person that draws red loses. Before any players turn they can choose to draw from the deck as is, or cut the deck. A cut consists of removing anywhere between 1 and number of remaining cards minus 1 cards from the top of the deck and placing them underneath the remaining deck.

The first person goes and draws a black without cutting. What are the probabilities the next person loses if they decide to cut (or not cut ) the deck before their draw?

@joe-shmo

After player 1 draws a black card, there are 20 cards left in the deck.

There are 18 ways to mix in the red cards. If player 2 does not cut, the odds of pulling a red card are 1 in 18. (The case with 3 red at the top.)

1/18 = 0.0555...

There are 19 ways to cut the deck. In the case of 3 reds on top, cutting results in a 2/19 chance of drawing a red. For the other 17 cases, there is a 3/19 chance of drawing a red.

Total all this up, and there are 3*17+2 = 53 ways to draw a red, after cutting, and 18*19 = 342 possible outcomes of cutting.

53/342 ~= 0.155

Player 2 should NOT cut the cards.

@joe-shmo said
If they should cut, nobody knows....or do we?

Two people are playing a card game with 18 Black cards, and 3 red cards. The red cards are randomly mixed into the black cards as a group ( the red cards are all together somewhere in the deck at the start of the game example of possible configuration: BBBBBRRRBBBBBBBBBB )

The game is played by drawing card ...[text shortened]... probabilities the next person loses if they decide to cut (or not cut ) the deck before their draw?

@bigdoggproblem said
@joe-shmo

After player 1 draws a black card, there are 20 cards left in the deck.

There are 18 ways to mix in the red cards. If player 2 does not cut, the odds of pulling a red card are 1 in 18. (The case with 3 red at the top.)

1/18 = 0.0555...

There are 19 ways to cut the deck. In the case of 3 reds on top, cutting results in a 2/19 chance of drawing a re ...[text shortened]... d 18*19 = 342 possible outcomes of cutting.

53/342 ~= 0.155

Player 2 should NOT cut the cards.

@venda said
Steady on Joe,one at a time mate.(I believe the correct expression is lol-(I don't use these things or the silly faces)
So to begin, including a question at the end.
Let's call the red cards r1 r2 and r3 and 14 black cards after one has been drawn
The probability of a red card being the top card is 1:15 =.066.(It can only be r1)
With a cut it becomes a lot more complicated:-

@joe-shmo
Yay / whew.

@bigdoggproblem said
@joe-shmo
Yay / whew.

@joe-shmo
I did! Started with the math this time, instead of brute-forcing it.

@joe-shmo said
"Steady on Joe,one at a time mate."

In my defense I did post this yesterday!

"Let's call the red cards r1 r2 and r3 and 14 black cards after one has been drawn
The probability of a red card being the top card is 1:15 =.066.(It can only be r1)
With a cut it becomes a lot more complicated:-"

Correction, they start with 3 red 18 Black. After the firs ...[text shortened]... s, its a random cut. After the cut is complete there may be black cards in between red cards.