Hello all. Please forgive me if I do not post in accurate mathematical language. My last math was a beginning calculus course in 1979. My first dilemma is whether to post irrelevant background. I shall do so. Please skip to paragraph 3 if you do not care.
My son (5th year engineering senior at OSU) and I have been having a discussion lately about a game my Dad told me about from when he was a POW. He was shot down over Austria in 1943 and spent about 18 months locked up. Not much to do in an enemy prison camp, talk about food, think about food, dream about food. Substitute "girls" for "food". Repeat. Anyway, they did have cigarettes and cards. One of the things they did was a gambling game with the cards. The player pays one cigarette and takes the deck from the house. Turning over the top card the player counts "one." As long as the turned over card is not an Ace, the player continues. He turns over the second card and counts "two." As long as the turned over card is not a two, he continues, 1 through 13 and repeats 4 times. As long as he never turns over a card equal to his count, he wins and the house pays. Here is question part. My Dad told me house paid 25 to 1. I tried to convince him that he was being ripped off at that rate and he should always be the house. He insisted that he saw thousands of iterations of this and those odds were right on. I tried to convince him that there was chicanery involved, perhaps paying out too frequently to a confederate to encourage playing at unfavorable odds. I never convinced him, but never really new the right math to do it anyway.
So what is the probability of success for counting sequentially through a deck of cards (1-13 in order, repeat 4 times) and never turn over a card equal to your count. Early on I thought the odds were less than .01 but I progressed to where I thought it was (12/13)^52 or about 64 to 1. My son built a simple brute force excel spread sheet that after a few thousand iterations seems to suggest slightly longer odds around .0137. So what is the mathematically correct way to figure this out?
Thanks for you attention.