brilliant.org概率论导论学习笔记。 先看概率论，基本说就是college student以下的内容，加快速度。 但是需要一个月30-60刀的订阅，算了，还没到这个时候。
Probability gives a nearly universal framework for analyzing the probabilistic world around us, with applications that stretch from games to sports to finance to engineering to medicine. For example, as later problems will show, probability can help us determine our confidence that someone committed a crime given some evidence in the case.
Thinking Probabilistically
Try again
Which of these events is the most likely to happen when flipping a fair coin?
Flip 2 or more heads when flipping 3 coins.
Flip 20 or more heads when flipping 30 coins.
Flip 200 or more heads when flipping 300 coins.
Instead of making an explicit calculation, think about how “surprising” each of these outcomes would be.
好比binomial distribution， 当发生较大的大数定理时，\(\mu\)不变，\(\sigma \downarrow\)，越集中，因此\(2/3\times N\)分的概率越来越小，长尾风险越来越小。因此\(p \downarrow\)。 非常好的一个sense题。
It is not clear that SIDS (which could be caused by some genetic predisposition) is actually independent across children from the same parents.
The “a priori” probability that Sally Clark would commit a double homicide was also incredibly rare, if not more rare than double-SIDS.
Note: “A priori” refers to the probability that Sally Clark would commit such a crime before taking into account the information of the deaths, which is the same type of probability that Sir Roy Meadow calculated. You can assume the facts are true, and you are simply evaluating reasonableness of the arguments.
这里讲的就是置信区间的问题。
Approximately how many times more likely would an observation of 2-sigma significance be than an observation of 5-sigma significance, assuming that the observation happens by chance?
在讲置信度，
- \(2\sigma\to0.05\),
- \(5\sigma\to3\times 10^{-7}\)
\(\to\) \(0.15\)million times.
In WWII, enemies would engage in plane-to-plane aerial combat. Unsurprisingly, many planes were lost to crashes; if a bullet strikes a plane in a sensitive area, it’s very hard to make it back to base. For the planes that did come back, the mechanics kept track of the location of bullet holes in the fuselage, so that they could reinforce the planes in the most vulnerable locations.
For American planes, the bullet holes on returning planes were distributed as follows:
(Assume all parts are shot at with roughly equal frequency.) Where should the mechanics reinforce planes so that more of them come back safely?
这里并不是考虑降低\(P(wings)\)的概率， 而是一个条件概率， \(P(wings|back)\)大并不代表wings脆弱。 因为很可能\(P(engine|not \space back)\)非常大。 最好导致了\(P(engine) = P(engine|not \space back) \uparrow \uparrow + P(engine|back) \downarrow = \uparrow\)。 这是一个思维的漏洞。 \(P(engine,back) = P(engine|back)\cdot P(back)\)。
In a certain game of tennis, Alex has a 60% probability to win any given point against Blake. The player who gets to 4 points first wins the game, and points cannot end in a tie.
What is Alex’s probability to win the game? Try to use your intuition, rather than making a calculation.
\(> 60%\)
这又是一个binomial distribution的问题。 理论上Alex打赢4个球就赢了，且结束。假设他们打了\(N \geq 4\)盘。 当\(N \uparrow\)， 打赢\(\mu = 0.6N\)的概率不变，但是打赢\(\leq4\)球的概率越来愈小，理论上，\(\geq 4\)的概率越来越大，因此必然大于60%。 这个时候，也反映了，当敌强我弱的时候，一定要速战速决。