What is the probability that the sun will rise tomorrow?

1 min read

Laplace introduced the sunrise problem in 18th century. With prior knowledge that the sun has risen N days in the past consecutively (not knowing the gravity rules etc), what is your confidence that the sun will rise tomorrow?

Let p be the probability that the sun rise. Apparently p range from 0 to 1. Without any prior knowledge, we have same level of confidence to believe p=0, 0.1, 0.2, …, or 1. So we assume equal probability distribution on p.

Now assume day 1 passed and we found the sun rose. Based on this knowledge, how should we update our belief on p? Well, p=0 is impossible, otherwise we will have not observed the sunrise. p=0.1 is possible but unlikely. Because if p=0.1 we only have 10% probability to observe a sunrise but we did observe one. We can actually formally calculate P(p=q|N=1) = P(N=1|p=q) * P(p=q) / P(N=1) using Bayes’s theorem. Since p(N=1) is in the denominator and it doesn’t depend on q, we don’t care. Our prior belief of P(p=q) was a uniform distribution. P(N=1|p=q) is equal to q. So the result is P(p=q|N=1) = q. After normalization (i.e. our total ‘belief’ over all p should be 1), P(p=q|N=1) = 2*q. So we have updated our belief from uniform distribution to a skewed distribution towards bigger probability of sunrise.

Based on our new belief (i.e. P(p=q)=2*q), what is our confidence that the sun will rise tomorrow? We simply need to sum (or integrate) all the possibilities. The final result equals to int(2*q^2) = 2/3. Yes, we have 2/3 confidence to say the sun will rise tomorrow, given we have observed one sunrise.

We then continue doing this for N=2, 3, …. The confidence level equals to (N+1)/(N+2). If we have observed sunrise for 1000 days, our confidence level raised to 0.999.

The belief of the probability of sunrise based on n observations of sunrise
The belief of the probability of sunrise based on n observations of sunrise

But if we have prior knowledge of the gravity rules, our belief will be dramatically different.


Want to receive new post notification? 有新文章通知我


两千多年前的战国时期,魏王的大臣庞葱对魏王说:“如果有人说外面的街市有老虎,您相信吗?”魏王说不信。“如果又有一个人说呢?”魏王说:“我将半信半疑。”“如果有第三个人说呢?”魏王说:“我会相信。” 上
Xu Cui
26 sec read

LaTex support (2020)

I used to use jsMath to support latex in this website, but it no longer works (I have no idea why). The one I am using right now in this blog is MathJax LaTex WordPress plugin. A pleasant surprise is that my old equations (e.g. in Balloon model) stil
Xu Cui
29 sec read

Recommend 3blue1brown

When I was in high school and saw the following equation, my mind was blown! $$1+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+…=\frac{\pi^2}{6}$$ Why is Pi here? Isn’t it supposed to show up only in circle related problem? But the left-hand has n
Xu Cui
22 sec read

One Reply to “What is the probability that the sun will rise…”

  1. If n observations is equal to the number of days the world has existed what is the probability that the sun will rise? This with the prior knowledge of gravity, what is the probability, is it known or solvable?

Leave a Reply

Your email address will not be published. Required fields are marked *