Probability involving a deck of cards

[karthikphenom]

Homework Statement

A deck of 52 playing cards are tosses 1 by 1 into a large box containing 4 sub-compartments. On a given throw, a card is equally likely to fall into any of the sub-compartments. After all 52 cards are thrown into the box, what's the probability of getting these distributions
a) 13-13-13-13
b) 16-22-10-4

I'm bad at doing probability calculations, and I have no idea of how to proceed. Can someone please help me out?

 

[lanedance]

pretty tricky.. and there might be an easier way i haven't though of, but this should at least get you started thinking about it

start with one card what are the possible outcomes & probabilties for each, then try 2 & 3 to get a feel for the problem

then i haven't tried this but then i would consider one box, what is the probabilty it ends up with "i" cards after the full 52? could use a binomial distribution...

those "i" cards are essentially independent from which other box the others land in, so given (52-i) what's the probablilty the next box gets "j" cards and so on

then consider whether the order of the boxes matter...

you can probably work out the total number of outcomes using something like combinations with repetition

 

[Architeuthis Dux]

I understand the importance of probability in analyzing and predicting outcomes. In this scenario, we are dealing with a deck of 52 cards being randomly distributed into four sub-compartments. The probability of a card falling into any one of the sub-compartments is equal, as stated in the problem. Therefore, we can use the basic principle of probability to calculate the likelihood of getting specific distributions.

a) For the distribution of 13-13-13-13, we can calculate the probability as follows:

Probability = (Number of ways to get desired outcome) / (Total number of possible outcomes)

Number of ways to get desired outcome = (Number of ways to choose 13 cards from 52) * (Number of ways to choose 13 cards from remaining 39) * (Number of ways to choose 13 cards from remaining 26) * (Number of ways to choose 13 cards from remaining 13)

= (52C13) * (39C13) * (26C13) * (13C13)

= 1.11 x 10^26

Total number of possible outcomes = (Number of ways to distribute 52 cards into 4 sub-compartments)

= 52^4

= 7.311 x 10^9

Therefore, the probability of getting a distribution of 13-13-13-13 is:

Probability = (1.11 x 10^26) / (7.311 x 10^9)

= 1.52 x 10^16 or approximately 0.0000000000000152

b) Similarly, for the distribution of 16-22-10-4, we can calculate the probability as follows:

Probability = (Number of ways to get desired outcome) / (Total number of possible outcomes)

Number of ways to get desired outcome = (Number of ways to choose 16 cards from 52) * (Number of ways to choose 22 cards from remaining 36) * (Number of ways to choose 10 cards from remaining 14) * (Number of ways to choose 4 cards from remaining 4)

= (52C16) * (36C22) * (14C10) * (4C4)

= 3.22 x 10^22

Total number of possible outcomes = (Number of ways to distribute 52 cards into 4 sub-compartments)

=