# Probability Musing

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Please let me know if you know the solution to the following problem:

What is the probability of me waking up at 10am?

What additional information should be supplied so as to determine the probability? What do you exactly mean by the probability of this event? Which kind of conditional probability will make sense?

Consider the following comment by Timothy Gowers regarding the model for calculating the probability of an event involving a pair of dice:

Rolling a pair of dice (pp. 6), Mathematics: A very short introduction Â© Timothy Gowers, 2002 [Source]

I find probability very confusing, for example, this old post.

### 16 responses

1. I hope this helps…probabilities deals with “laws of averages”…it is axiomatic as laid down by A Kolmogorov. You can refer to the preface and introduction chapter of the book, Probability: Random Variables, and Stochastic Processes by A. Papoulis, et al. An Indian edition is available…Hope this helps…

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• Ok. There are lots of flaws.

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2. That’s another problem in itself. Is there an instant of time? And how can you exactly measure it? And I mean exactly. You would need to solve that problem first.

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• Yes, time is relative. Time would be running at different rates at different points of the body. Hence 10 am is vague.

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• But we don’t know how to model it, so as to make predictions reliable. For example, according to a popular model Riemann hypothesis is true by probability 1. But it doesn’t help in any way.

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• Ok. Assume I was sleeping till 10 am

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• It depends then on whether you set an alarm. Whether you set it at the correct time and the device is reliable. It also depends on what you call waking up. If you automatically switch it off when it rings and do not actually rouse, it is a moot point whether you ‘woke up’ or not.

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• I don’t set alarm. I am randomly waking up. Waking up means eyes open and well aware. Since time is continuous, it’s like the probability of a pin falling inside a circle (like Buffon’s needle experiment). But should we consider 24hrs or 86400sec etc.

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• This is a real world problem. It’s not abstract. For example, if you wake up on the hour it is always 10am somewhere in the world. Time is relative. Travelling at high speed through space, for example, alters the measurement of time. Also, which 10am do you mean? The next available 10am, the one after, or the one after that…? In the real world, probability becomes meaningless and undefinable. Even the probability theory of a real tossed coin is meaningless because a real coin is not absolutely perfect and uniform.

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• Ok. But what would be the simplest model? I think an instant of time has probability zero over whole interval

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• If you wake up randomly, and if time is continuous(it could also be discrete) then isn’t the probability of you waking up at 10:00, zero?

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• Yes. This is what I think

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