| Title | Envelope Trade-Up |
|---|---|
| Contributor | David H. Silver(author) |
| DOI | https://doi.org/10.11647/OBP.0526.15 |
| Landing page | https://www.openbookpublishers.com/books/10.11647/obp.0526/chapters/10.11647/obp.0526.15 |
| License | https://creativecommons.org/licenses/by-nc/4.0/ |
| Copyright | David H. Silver |
| Publisher | Open Book Publishers |
| Published on | 2026-04-08 |
| Long abstract | The Envelope Paradox presents two envelopes where one contains twice the money of the other. After selecting one envelope, seemingly valid probabilistic reasoning suggests an expected gain by switching (averaging x/2 with 2x), regardless of which envelope was initially chosen. This symmetric conclusion creates a logical inconsistency since perpetual switching cannot be optimal. The paradox arises from improper application of expected value calculations to scenarios with unbounded distributions or when conditional probabilities are not properly accounted for. Resolving the paradox requires distinguishing between known values and variables, recognizing when probability distributions are ill-defined, and understanding the limitations of calculations with potentially infinite quantities. |
| Print length | 10 pages |
| Language | English (Original) |
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| Landing Page | Full text URL | Platform | |||
|---|---|---|---|---|---|
| https://www.openbookpublishers.com/books/10.11647/obp.0526/chapters/10.11647/obp.0526.15 | Landing page | https://books.openbookpublishers.com/10.11647/obp.0526.15.pdf | Full text URL |
David H. Silver is an industrial researcher whose career bridges computer vision, computational biology, and science communication. He studied mathematics, computer science, and biology at the Technion — Israel Institute of Technology as a Rothschild Scholar, and was awarded a Microsoft Research PhD Fellowship for his doctoral work in computational biology at Cambridge, UK. Silver’s peer-reviewed publications span multiple domains: computational biology in Nature and PNAS; computer vision systems in IEEE Transactions on Pattern Analysis and Machine Intelligence; medical AI in Human Reproduction and MIDL; and entertainment analysis in PLoS One. He holds over a dozen patents in depth sensing, medical imaging, and generative AI. His industry positions include Algorithm Engineer at Intel Corporation, ML Researcher at Apple, and CTO/co-founder roles at several technology startups. Silver maintains academic collaborations with researchers worldwide and serves as a peer reviewer for Image and Vision Computing and PNAS.