- #1

murshid_islam

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- TL;DR Summary
- Probabilities of winning in a 3-person card game

Suppose I have a shuffled pack of cards, out of which 4 red cards have been lost or removed. So now I have 22 red cards and 26 black cards in random order. I'm dealing cards in pairs. If a pair has 2 red cards, they are dealt to Mr. Red, if the pair has 2 black cards, they're dealt to Mr. Black, and if the pair contains 1 red and 1 black card, they are dealt to Mr. Mismatch.

After all the cards have been dealt in pairs, whoever has the most cards wins. Now, Mr. Mismatch will almost always win because there are more ways to get a mismatched pair (card-1 is red and card-2 is black or vice versa) than a pair of reds or blacks. Also, since there are 4 fewer red cards, Mr. Red will never win.

I want to know the probabilities of Mr. Black and Mr. Mismatch winning. I already ran a simulation (in Python) 1,000,000 (one million) times to see what percentage of games are won by the three players. Here's my code in case anyone wants to check it out

This is the result I got:

Mr. Red won = 0 times = 0.00%

Mr. Black won = 93799 times = 9.38%

Mr. Mismatch won = 906201 times = 90.62%

No matter how many times I run my code, I get similar results: Mr. Black winning little more than 9% and Mr. Mismatch winning a little more than 90%. However, Someone claimed that Mr. Mismatch should win about 95.6% of the time. I was wondering if it's my code that is wrong. So, I wanted to calculate the probabilities using math to check if my results are correct or not. Any ideas on how I can calculate the probabilities of their winning?

After all the cards have been dealt in pairs, whoever has the most cards wins. Now, Mr. Mismatch will almost always win because there are more ways to get a mismatched pair (card-1 is red and card-2 is black or vice versa) than a pair of reds or blacks. Also, since there are 4 fewer red cards, Mr. Red will never win.

I want to know the probabilities of Mr. Black and Mr. Mismatch winning. I already ran a simulation (in Python) 1,000,000 (one million) times to see what percentage of games are won by the three players. Here's my code in case anyone wants to check it out

This is the result I got:

Mr. Red won = 0 times = 0.00%

Mr. Black won = 93799 times = 9.38%

Mr. Mismatch won = 906201 times = 90.62%

No matter how many times I run my code, I get similar results: Mr. Black winning little more than 9% and Mr. Mismatch winning a little more than 90%. However, Someone claimed that Mr. Mismatch should win about 95.6% of the time. I was wondering if it's my code that is wrong. So, I wanted to calculate the probabilities using math to check if my results are correct or not. Any ideas on how I can calculate the probabilities of their winning?