Est. 1998 — 27 years of woodworking knowledge
Tuesday Puzzle
Here is a very easy one to see if we can get some of the lurkers to come out ;-)
I have a bag that contains a solid color billiard ball. I select another ball from a supply of balls in such a way that I have no idea whether it is a solid or a stripe and place it in the bag. I then draw a ball from the bag. If it is a solid, what is the probability that the other one in the bag is also a solid?
Worthless hint
Think of what Tampa, Mobile, and San Francisco have in common.
And if you get what that has to do with the puzzle, I'm sure you have already solved it!!
Answer
There are four possible outcomes, all equally likely.
* Solid added; original solid drawn.
* Solid added; new solid drawn.
* Stripe added; original solid drawn.
* Stripe added; new stripe drawn.
Since we know a solid was drawn from the bag we can exclude the final outcome. In two out of the three remaining outcomes the other ball is a solid.
Therefore the probability that the ball remaining in the bag is also a solid is 2/3.
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Despite my hint about cites with Bay[e]s, I approach such problems from first principles, exactly as you did.
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My daughter gets mad at me whenever I help her with her math because she doesn't like it when I want to go back to the basics: What are they asking for?, What info do I need to get it?, What do the pieces in the equations actually mean?, Do I have all of it? If not, can I derive it or get it some other way? ...... Funny thing. One time I was helping another daughter solve a problem and ended up with a solution equation that worked. Later, I was doing a bit of looking up and it turned out that I had derived the law of cosines, which I had forgotten ;-) Now, if I had remembered it, the solution would have gone faster, but since I didn't remember, if I had relied on memory, the problem would not have gotten done.
I always hated memorizing formulas. If I misremembered, I blew the whole thing. I majored in EE and memorized two formulas: E=IR and P=IE. With those and some basic techniques, you can derive most anything else you need.
I admit to being fairly deficient in statistics. I know enough to get only slightly dangerous.
This is part of why I enjoy the trivia and puzzles. It helps keep my mind (such as it is ;-)) active and I almost always end up learning something thanks to you and all that play here.
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Sounds like me!
I was a math major, but even as far back as jr high, I was big on understanding, not memorizing. Almost got me into trouble with trig in high school. Similar to your EE attitude, I refused to memorize trig identities, saying that if I couldn't figure it out from the definitions and sin^2 + cos^2 = 1, then it was probably not worth knowing!
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I was an art major (on paper at least). The only thing we had to memorize were thousands of art history slides.
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I feel for you. At least it is possible to go back to basic principles in math. In your case, brute force memorization would be the only alternative. I can remember my name most of the time, but memorizing that much stuff might overtax what brain I have left ;-)
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The memorization part wasn't fun and I have forgotten much of the detail. Nevertheless, it was worthwhile to have been exposed to it for the historical perspective.
There are no 'basic principles' of art per-se, but there are many techniques, devices, conventions and formulas that have been invented, discovered, borrowed, derived, forgotten and re-discovered over the centuries. Some of them actually quite mathematical in nature.