As a mother of a 15-month old I have not slept straight through the night for weeks or months. Perhaps even years. My memory is so weakened by this lack of sleep that I don't even remember:) February colds with caught and sneezes, belly ache, teething - all make their way to the sensitive mommy's ears waking me up a few times each night. Every time I am awaken I barely remember who and where am I and whether this is the first time I am up this night or second. So, here comes the puzzle:

You are going to sleep and you know that every night you will be awaken either once or twice, chances of each are equal. Let's say a flip of a coin by a sleep angel decides: heads - you will be awaken once, tails - twice. You will be so tired that after waking up you will go back to sleep and you won't remember this awakening later on. Now, I have two questions for you:

- On any of these awakening, what do you think are the chances that the original coin showed heads?
- Any time you wake up what are the chances that this is your first and only time awake this night?

This puzzle was invented by Arnold Zuboff and goes under a slightly different name which I will reveal later.

The sleeping mom and baby image by NoVa Hokie, distributed under CCL.

Your answers accepted any time until midnight on Sunday on our Family Puzzle Marathon. They will be hidden till then and everyone who submitted something reasonable will get a puzzle point.

## 15 comments:

There is a 1/3 chance on any awakening that the coin showed heads. The first awakening would be 50/50 and the second awakening would be 0/100. SM doesn't know if it is the first or second so the chances are .5/1.5 or 1/3 in favor of heads.

As for 1st or second awakening, I believe there is a 50/50 chance. She either wakes once or twice and these probabilities are independent of one another.

On any two evenings, the expected number of wakenings is three (one night with one wakening, one night with two).

(1) On one of those nights, the coin was a head, the other a tails. You wake three times, one of them had a head, so 1/3 of the wakenings correspond to an original coin that was a heads.

(2) This is 1/3 as well - you wake up three times, only one of them first and only.

I have a sneaking suspicion I am over-simplifying this, but this is what my intuition says...

-Kim

That's a tricky one! Let's see.

"Every night you will be awaken either once or twice, chances of each are equal" (text)

Let's pick a day: Sunday.

P(Tails|once on Sunday) = P(Heads|twice on Sunday)but chances of each are equal so, we have:

P(Tails and one on Sunday) = P(Tails and twice on Sunday) = P(Heads and once on Sunday).

The probability of each should be one-third.

However,my first reflex without thinking (Harvard problem) would have been to say 1/2.

No matter the day, each time this person awakes, only one-third of the cases with heads would precede her action. I would expect the same probability P(Heads)=1/3.

1. Flipping the coin, the chances of Heads is 50%, just like Tails. Once per night.

2. Over many nights, I'd waken an average of 1.5 times per night. Either "1" or "1,2." It's tempting to say there will be more "1" awakenings, but that's a trap. The puzzle says "FIRST AND ONLY time awake this night," and the chances of that will also be 50%.

Q.#1 If you toss the coin there is 1 chance out of 2 possible outcomes that the coin will land on the "Heads" side.

Q.#2 If you awaken during the night and the coin face shows either heads or tails, the chances of the 2 possible outcomes will be the same; there is 1 chance out of 2 possible outcomes that the sleep angel tossed either heads or tails.

I think it is 50% for both questions.

Gurubandhu

Question 1:

The chance that the original coin toss showed heads is: 1/2

The part of the question that threw me a little is the "on any of these awakenings" what is the chance that the first coin was heads?" I think the answer is still the same though. There is only one coin toss, and an equally likely chance of getting heads (50%) or tails (50%).

Question 2: The chance that any time you wake up, it will be your first and only time awake: 1/3.

There are 3 times you could wake up:

You draw H and wake up one time

You draw T and wake up the first time

You draw T and wake up the second time

Only in the first case are you only waking up once, which is why 1 of 3 is my answer.

Sleep: I am sorry that your youngest child is keeping you up so much and preventing you from getting a good night's sleep. My 4-year old was such a bad sleeper when he was younger -- I don't he slept through the night regularly until he was at least 2; and even still when he gets into bed with me he sleeps so restlessly and moves so much, any chance of me getting a good sleep for the rest of the night is nil. A while back us 2 parents made a deal with him, if he slept the whole night in his own bed, he would get a quarter in the morning (a good night's sleep is worth at least a quarter, right?! :)). This has been a great incentive for him, though sometimes when he wakes up in the night he comes and asks if I'll join him in his bed, because that way he'd still eligible for the quarter. In addition to more nights with better sleep, an bonus of this arrangement is that he has because quite proficient at counting change and exchanging quarters for dollars and so on.

Good luck with your sleep situation. Since my 4-year old was born, I have become adept at falling asleep quickly whenever given the chance....and I don't even need to lie down to do so. I am sitting watching TV and then boom, I'm asleep. It can be a little disconcerting, but at least my body is helping me find a way to get the hours of sleep I need.

take care,

TracyZ

One has to decide whether the problem is about the coin or the person.

If the problem is about the coin, the answer is obvious. It is 1/2. The consequences of what happens after the coin is tossed has nothing to do with the coin. The consequences are independent or man made. The coin itself is said to be fair, hence 1/2.

But if one looks at you the mother, and considers the consequences of the coin toss, then there are 3 likely outcomes.

Heads: you will be awakened once and only once.

Tails(1): you will be awakened twice and this is your first awakening.

Tails(2): you will be awakened twice and this is your second awakening.

Since all three are equally likely the answer for it being heads is 1/3.

Personally, I think this problem is mixing apples and oranges. Consequences should never be mixed with the mechanism producing them. I go with the 1/2.

Consider what would happen if you expanded the problem to tails meaning that you would be awake 3 times. Does the probability of it being heads go down to 1/4? Does this seem reasonable?

Or does it seem more reasonable to say that the consequences of throwing a heads is 1/4?

I just thought about my answer and the tone seems a little gruff.

I don't mean it that way. The problem is a terrific problem. It is only that the pragmatic part of me has a very strong opinion (which is perhaps incorrect).

1. I think the chances for the coin going heads on any of theses awakening is 1/2.

2. I think the chances of being your first and only time is 1/3.

Given the equal probability of heads and tails, a large sample of 2*N nights will consist of roughly N heads (and single awakenings) and N tails (double awakenings). That amounts to total 3*N awakenings. So here is the reasoning for the answers:

1. I see two interpretations of the question, around the meaning of “original coin”:

1.a. When you say “original coin”, you are referring to the coin tossed that night. The chances should be calculated by taking the number of the specific outcomes and dividing by the total number of outcomes. Out of the total of 3*N awakenings, only N will have happened on a night with the coin showing heads, so the probability is 1 / 3.

1.b. When you say “original coin”, you are referring to the very first coin toss, or maybe what the coin showed even before it was tossed for the first time? If this is the case, then of course the probability is 50% because all the awakenings have nothing to do with this original single state.

2. I see no difference between this and 1a. Awaking for the first and only time will happen when and only when the coin toss on that night resulted in heads. So the answer is the same as for 1a: 1 / 3.

Both questions have an answer of 1/3.

Since heads and tails are equally likely, there is an equal chance of one or two awakenings. Of those three awakenings, one of them is on a heads night, the same one of the 3 that represents the first and only awakening of a night.

You all felt right that both questions refer to the same event and therefore their answers must be the same. 1/2 or 1/3 but the same. If you Google this puzzle as "The sleeping beauty problem" you will see various mathematical and philosophical arguments for either of the answers!

I personally like the 1/3 answer more. Here is why:

In any coin toss chances of heads or tails are equal 1/2. Imagine a group of kids that each throw a coin and scream either "heads" or "tails" depending on how each coin falls. You would expect to hear as many "heads" screams as "tails". Now tell each kid that if his coin falls "heads" he should scream once but if his coin falls "tails" he should scream twice. You would then hear twice as many "tails" screams as "heads" screams. Therefore, chances of "head" scream are becoming 1/3.

This is just my analogy. Each of you provided beautiful arguments for your solutions, so read it all above carefully. Puzzle point to everyone who posted.

And thank you so much for your support in my lack of sleep. I know it is only temporary. TracyZ - love your quarter idea. Our neighbor Peter warned us when we decided to have a third child - he told us that that the amount of work and exhaustion is not increasing by 50% (2 kids to 3 kids) but at least by 100% But they are well worth it!

A few more things -

Katrina it is so nice to see you back. To those who do not know Dennis and Katrina are one of the husband-and-wife teams that used to submit puzzle answers together then started competing. Katrina disappeared for a few months and Dennis I think wrote that you both had a fourth baby. You should know even more than I about lack of sleep.

TyYann - I am certain that I saw you before many times in this puzzle blog but by some reason I do not see that you got any puzzle points in the list on the left. Perhaps I screwed up and accidentally erased them. Let me know if your points are indeed missing and we will re instantiate them.

have a warm week!

Thank you Maria. It is actually our 5th but at this point, he is rarely the cause of our sleeplessness - it is more often the almost teenager. Interesting puzzle and as a mom of 5, my best sleep advice is to do whatever gets the most people the most sleep. I also don't track awakenings by the baby because knowing how often or long he has me up makes me feel more tired.

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