Probability Puzzles

Problem Statement

There are 50 cards of 5 different colors. Each color has cards numbered between 1 to 10. You pick 2 cards at random. What is the probability that they are not of same color and also not of same number?

Solution

There’re 50 * 49 ways to select 2 cards from the deck – it’s the total number of possible outcomes. Now we need to calculate the number of positive outcomes: we can pull the first card in 50 ways, once it’s done, the second card should have a different color and a different number, so out of 49 cards left only 49 - 9(same color) - 4 (same number) = 36 will satisfy our criteria.

Answer: P = 50 * 36 / (50 * 49) = 0.734

Not convinced? Try the simulation below!

Simulation

show / hide simulation code


import random
from collections import namedtuple

nrounds = 1000
positive_outcomes = 0

deck = []
card = namedtuple('card', ['color', 'number'])
for color in ['a','b','c','d','e']:
    for n in range(1,11):
        deck.append(card(color, n))

for i in range(nrounds):
    pair = random.sample(deck, 2)
    first_card = pair[0]
    second_card = pair[1]

    if first_card.color != second_card.color and first_card.number != second_card.number:
        positive_outcomes += 1

simulated_probability = round(positive_outcomes / nrounds, 4)