card_game.py

# Draw 5 cards from the standard deck of 52 cards. Denote random variables X to be the number of black cards in the hand and Y to be the number of spades in the hand.

from math import comb
from pandas import DataFrame

nways = comb(52, 5)

joint_pmf = [[0.0 for x in range(7)] for y in range(7)]

total = 0.0

# populate the joint_pmf
for y in range(6):
    for x in range(y + 1):
        joint_pmf[y][x] = comb(13, x) * comb(13, y - x) * comb(26, 5 - y) / nways
        total += joint_pmf[y][x]

print(f'total probability for p_XY is {total}')

EX = 0
total = 0.0

# calculate marginal pmf for X
for x in range(6):
    px = 0.0
    for y in range(6):
        px += joint_pmf[y][x]
    joint_pmf[6][x] = px
    EX += x * px
    total += px

print(f'total probability for p_X is {total} and expectation is {EX}')

EY = 0
total = 0.0

# calculate marginal pmf for Y
for y in range(6):
    py = 0.0
    for x in range(6):
        py += joint_pmf[y][x]
    joint_pmf[y][6] = py
    EY += y * py
    total += py

print(f'total probability for p_Y is {total} and expectation is {EY}')

print(DataFrame(joint_pmf))

covXY = 0
for y in range(6):
    for x in range(y + 1):
        covXY += (x - EX) * (y - EY) * joint_pmf[y][x]
print(f'Cov(X,Y)={covXY}')

EXY = 0
for y in range(6):
    for x in range(y + 1):
        EXY += x * y * joint_pmf[y][x]
print(f'Cov(X,Y)={EXY - EX * EY}')