Article #010: Sleeping Beauty Problem

My point of view towards the sleeping beauty problem as a halfer.

Problem Statement

Sleeping Beauty volunteers to be the subject of an experiment. And before it starts, she is informed of the procedure: on Sunday night she will be put to sleep, and then a fair coin will be flipped. If that coin comes up heads, she will be awakened on Monday and then put back to sleep. If the coin comes up tails, she will also be awakened on Monday and put back to sleep, but then she will be awakened on Tuesday as well and then put back to sleep. Each time she gets put back to sleep, she will forget that she was ever awakened. In the brief period any time she is awake, she will be told no information, but she will be asked one question: "What do you believe is the probability that the coin came up heads?"

If you think, the answer is 1/2, you are named a Halfer. But if you think the answer is 1/3, you are named a Thirder. And if you think the answer is something else, you are named a Conspiracy Theorist.

Opinion

I am a halfer. I think the answer to "What do you believe is the probability that the *fair* coin came up heads?" is obviously 50%. I don't understand how a fair coin can have the odds of coming up heads anything other than 1/2. But if I was to choose what the face of the coin was when I wake up, or the what the name of the day is when I wake up, I would obviously choose Tails or Tuesday.

Here is a python script that shows that Tails and Tuesdays are more frequent:

    import random

    heads_monday_counter = 0
    tails_monday_counter = 0
    tails_tuesday_counter = 0

    # 10^5 is a large enough sample size to prove my point
    for i in range(10000):

        # 1 = Heads | 0 = Tails
        fair_coin = int(random.random()*2)

        if fair_coin:
            heads_monday_counter += 1
        else:
            tails_monday_counter += 1
            tails_tuesday_counter += 1

    output = "Monday Heads Count = " + str(heads_monday_counter) + "\n" +
             "Monday Tails Count = " + str(tails_monday_counter) + "\n" +
             "Tuesday Tails Count = " + str(tails_tuesday_counter) + "\n" +
             "Monday Count = " + str(heads_monday_counter+tails_monday_counter) + "\n" +
             "Tuesday Count" + str(tails_tuesday_counter) + "\n" +
             "Heads Count = " + str(heads_monday_counter) + "\n" +
             "Tails Count" + str(tails_monday_counter+tails_tuesday_counter) + "\n"

    print(output)
    

Here is a sample output that proves my point from the python script above:

    Monday Heads Count = 5086
    Monday Tails Count = 4914
    Tuesday Tails Count = 4914
    Monday Count = 10000
    Tuesday Count = 4914
    Heads Count = 5086
    Tails Count = 9828
    

Explanation

The script that automates the process of tossing fair coins and keeping track of counters, provides accurate statistical data. But, we don't even really need a script to automate this for us. Thirders use this to explain how Heads occur only 1/3 of the time and 1/2 as much as Tails. Even though Heads is less frequent, the probability is still 1/2. In fact, Heads is less frequent precisely because the probability of a fair coin landing Heads is 1/2. Since, Sleeping Beauty (SB) wakes twice every Tails, it is bound to have twice the frequency of Heads. If the probability of a fair coin landing Heads was anything other than 1/2, then Tails wouldn't be twice as frequent as Heads.

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Written by BooleanCube :]