Learning is tough
Learning is tough because, while a little knowledge that you're missing can often bridge large gaps, it is not accessible to you  for if it was, you would reach out
and grasp it. And no one else can provide it, because no one but you knows exactly what you know and don't know.
This is why we have few friends in our neighborhoods, just acquaintances and strangers.


Solitude is precious
We are trained within
our culture from the onset of our lives to see things a certain way.
This is our cultural conditioning, and we often adopt it without realizing it
as such.
Almost always, we use good logic and the truths found within this
network to make judgments, decisions, and establish values.
Sometimes, we allow ourselves to believe in truths found
outside this network, which may (or may not) lie in
contradiction to ones inside it.
If we do so,
we think of the experience
as one of irrationality or spirituality.
For some,
it is a spiritual experience
that stems from and is guided by an outside source.

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However, true happiness can be found by working on probability
problems.
I found this about 8 years ago in a GNU book called Chance
Probability. It's not an elementary problem, unless you make assumptions. This problem originally appeared in:
K. L. Chung, Elementary Probability Theory With Stochastic Processes, 3rd ed (New York:SpringerVerlag, 1979), p. 152

Pickwick’s Umbrella
In London, half of the days have some rain. The weather forecaster is correct 2/3 of the time, i.e., the probability that it rains, given that she has predicted rain, and the probability that it does not rain, given that she has predicted that it won’t rain, are both equal to 2/3. When rain is forecast, Mr. Pickwick takes his umbrella. When rain is not forecast, he takes it with probability 1/3.
Can you find (a) the probability that Pickwick has no umbrella, given that it rains. (b) the probability it doesn’t rain, given that he brings his umbrella.
Note: You may not assume that the probability that she predicts rain is 1/2
or that the probability that she predicted rain given that it rains is
2/3; however, you need that, so prove it first.
Hint: Show that for any two events R & F such that a = P(RF) = P(~R~F) P(F) = 1  (a +P(R)) , 1  2a a ≠ ½ and under the further constraints that place the RHS between 0 and 1 inclusive. When P(R) = ½ a drops out of the RHS and P(F) = ½ . 

