Thursday, March 4, 2010

Night Adventures

Assume that you have three little kids that share one bedroom. Each of the kids occasionally comes to your bed at night, running away from bad dreams. On any given night there is 30% chance for each kid to end up in your bed. When two kids come to your bed on the same night, it starts to be too crowded. You have to leave them there and go to one of their beds to catch up a few more hours undisturbed. You take the alarm clock with you when you switch beds, to make sure you are the first to wake up and have a few quiet minutes before the hectic morning erupts. You have been waking up in your kids' room so frequently that you are starting to wonder whether you should leave the alarm clock as well as your clothes there on the prior evening. Is your chance of staying in your room through the night smaller than waking up in your kids room?

Submit your answer on our Family Puzzle Marathon Be first to solve three puzzles and get a prize!

8 comments:

Maria said...

Phil - I had to remove your comment.
I post three puzzles every week (easy, medium and hard) and in order to encourage more people to participate and not let experts like you get all the medals, I allow only one puzzle per person. You just solved the Parking Dilemma puzzle and won a puzzle point.
So, this puzzle is still open.

Phil said...

Rules is rules... :P

Kim said...

Maria, are you having trouble getting a good night's sleep? :-)

Maria said...

Well, this is fiction. But inspired by my life and friends' stories. I admit to occasionally waking up in a different bed than the one I went to sleep in...

Kim said...

Can you let me and Phil in? Maybe your rule about solving more than one could have a time component?

Maria said...

You are right - let's give it until Friday.
Meanwhile - how about Toblerone Math from last week? A great excuse to buy ten Toblerones and claim that you are doing some math :) Yum!

sydandlogsmom said...

I think you'd still have a better chance to wake up in your own bed.

Maria said...

You all are right - even with all these night adventures, there is a higher probability that you will wake up in your own bed. Let's see the math.
You will stay in your bed all night long, if neither of the kids will come into your bed or if only one of the kids will come.

Probability of neither of the kids coming over is: 0.7 x 0.7 x 0.7 = 0.343
0.7 is from a 30% chance of him coming and 70% chance not coming. 70% is the same as 0.7. We multiply 0.7 as these are independent events and for them to co-occur we need to multiply their probabilities.

Probability of only one out of three kids coming is: 0.3 x 0.7 x 0.7 x 3 = 0.441
0.3 is for one kid who came, 0.7 for kids who didn't, we multiply by 3 as any of the three kids can come.

So, your probability of waking up in your own bed is 0.343 + 0.441 = 0.784 much higher than a chance.

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