Question

Simulate each part of Problem 1 [Playing Cards] using a Python program. 10,000 trials should be plenty to get good approximations for these conditional probabilities.

This is problem 1:

1 [Playing Cards] Two cards are randomly chosen without replacement from an ordinary deck of 52 cards. Let E be the event that both cards are Queens. Let F be the event that the Queen of Spades is one of the chosen cards, and let G be the event that at least one Queen is chosen. Determine the following probabilities: (a) P(E|F) (b) P(E|G)

Answer 1

Python Code:

import random
"""Let a deck is made in such a way that queens are present at position 1,2,3,4 and queen of spades is 
present at position 1, you can choose any other numbers but then you will have to change your program accordingly """
#Calculating P(E intersection F) in 10000 trials
success=0
for i in range(0,10000):
    a=random.randint(0,52)
    while True:
        b=random.randint(0,52)
        if b!=a:
            break
    if a==1 or b==1:
        if a>=1 and a<=4 and b>=1 and b<=4:
            success=success+1
E_and_F=success/10000

#Calculating P(E intersection G) in 10000 trials
success=0
for i in range(0,10000):
    a=random.randint(0,52)
    while True:
        b=random.randint(0,52)
        if b!=a:
            break
    if a>=1 and a<=4 and b>=1 and b<=4:
        success=success+1
E_and_G=success/10000

#Calculating P(F) in 10000 trials
success=0
for i in range(0,10000):
    a=random.randint(0,52)
    while True:
        b=random.randint(0,52)
        if b!=a:
            break
    if a==1 or b==1:
        success=success+1
F=success/10000

#Calculating P(G) in 10000 trials
success=0
for i in range(0,10000):
    a=random.randint(0,52)
    while True:
        b=random.randint(0,52)
        if b!=a:
            break
    if (a>=1 and a<=4) or (b>=1 and b<=4):
        success=success+1
G=success/10000
P1=E_and_F/F
P2=E_and_G/G
print("P(E|F)= ",P1)
print("P(E|G)= ",P2)

Output:

Now let us manually calculate those probabilities and verify the accuracy of simulation: