Poker

We are interested in the distribution of the sum of 3 random cards drawn from a shuffled deck.

Where Ace = 14, and J, Q, K = 11, 12, 13 respectively.

import numpy as np

card_values = [14, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]

deck = card_values * 4

num_simulations = 10**6

draws = np.random.choice(deck, size=(num_simulations, 3), replace=True)

sum_of_draws = np.sum(draws, axis=1)

expected_value = np.mean(sum_of_draws)
variance = np.var(sum_of_draws)

expected_value, variance

(23.999912, 42.052581992256016)

Idea:

Note for Xi ~ Unif(a,a+1,…,b), E(Xi) = (a+b)/2, var(Xi) = 1/12*((b-a)^2 + 2(b-a))

Each Xi ~ Unif(2, 3, …, 14)

E(Xi) = 8

Var(Xi) = (12^2 + 2*12) / 12 = 14

Let S = X1 + X2 + X3, it follows that

E(S) = 24, var(S) = 42

Our assumption is that each draw is independent of the others (in reality it’s not, but we assume we are drawing from a population large enough, just to simplify calculations)