2 decks of 16 cards each: 3 red, 13 green. Chance of any person getting 2 red cards?

Question

I have a probability question that's been bugging me (probably should take a stats class)

I have two custom decks of cards. Each deck contains 16 cards. Of those, 3 are red and 13 are green.

If 16 people each draw one card from each deck, what is the chance that any of those people is holding a red card from deck 1 and a red card from deck 2?

Also, what would be the chance that 2 people are holding 2 red cards? What would the chance of 3 people holding 2 red cards be?

Answer 1

Consider the 16 cards drawn from the first deck - they wind up among the 16 people. Now let us think of those people in a line, such that the three on the left have a red card.

So now we ask, how many arrangements are there of 16 items where 3 are red, 13 are green? That is just the "combination of 16 things taken 3 at a time" = C(16, 3) = 16! / (3! * (16-3)!) = 16!/(3! 13!) = 8 * 5 * 14

The chance that all three people hold red cards is therefore 1/C(16,3) = 1/(8 * 5 * 14) = 1/560

The chance that exactly two people both hold red cards is C(3,2) * C(13,1) / C(16,3) = (3 * 13) / 560. So the chance that at least two people both hold red cards is 40 / 560 = 1/14

Finally the chance that exactly one person holds two red cards is (C(3,1) * C(13,2)) / C(16,3) = 3 * (13 * 12 / 2) / 560 = 9 * 13 / 280. So the chance that at least one person holds two red cards is (6*3*13+40) / 560 = (234 + 40)/560 = 274 / 560 = 137 / 280

Answer 2

I don't think that this is as hard as I first thought. Since everyone takes a card from the first pack then probabilities don't come into it there. You only want the probabilities that the people who first get red also get a red from the second pack. Let n be number with double red.
P(n = 3) = (3/16)*(2/15)*(1/14) = 1/560
P(n = 2) = 3*(3/16)*(2/15)*(13/14) = 39/560
P(n = 1) = 3*(3/16)(*13/15)*(12/14) = 117/280 = 234/560
P(n = 0) = (13/16)*(12/15)*(11/14) = 143/280 = 286/560
As a check (1 + 39 + 234 + 286)/560 = 560/560 = 1
The 3 at the start of the second line is because it doesn't matter which person of the three doesn't get a red.
The 3 at the start of the third line is because it doesn't matter which person of the three does get a red.

Answer 3

Changes to get 1 red card from 1st deck and 1 red card from 2nd deck is 3/16*3/16

if you build tree diagram, changes 2 person get 2 red card will be 2/32

changes for 3 person get 2 red card is 1