What is the probability that this is a "Wheel of Fortune" machine?

Question

I recently took a HackerRank test for a Data Science position and got the following question wrong:

A casino has 10 slot machines. Eight of them are "Wheel of Fortune" themed, and two are "Jeopardy" themed. However, you do not know which machines are which. A "Wheel of Fortune" slot machine wins on 20% of spins which a "Jeopardy" machine wins on 50%. Say you play a round and are a winner. What is the probability that this is a "Wheel of Fortune" machine?

I used posterior probability to calculate this: WoF = Wheel of Fortune

J = Jeopardy

W = Win

L = Lost

P(WoF|W) = (P(W|WoF) * P(WoF)) / P(W)

P(WoF) = 80/100 = 4/5
P(W|WoF) = 20/100 = 1/5
P(W) = (20/100)*(8/10)+(50/100)*(2/10) = 13/50

P(WoF|W) = 8/13

I came up with the answer of 8/13. Can anyone help me identify what I did wrong?

Answer 1

I am not convinced you did anything wrong here. Having looked at the work provided I suspected the answer was correct, so I went ahead and made a quick simulation in Python to test the answer you provided (8/13 or ~0.6154). Note that this simulation assumes the probability of picking a machine to play is uniform across all machines.

import random

machine_list = [1,1,0,0,0,0,0,0,0,0]

def pick():
    p = random.randint(0,9)
    return machine_list[p]

def play():
    p = pick()
    if p == 1: #J
       w = random.randint(1, 2)
        if w == 1:  # win
            return 3
        if w == 2:  # lose
            return 4
    if p == 0: #WoF
        w = random.randint(1, 10)
        if w <= 2:  # win
            return 1
        else:  # lose
            return 2


winlist = []
woflist = []

for i in range(10000000):
    p = play()
    if p == 1 or p == 3:
        winlist.append(p)
    if p == 1:
        woflist.append(p)

print len(woflist) / float(len(winlist))

With the following output:

0.615330464637