Numbers

I've got interested in prime numbers lately. (You can use the Sieve of Eratosthenes to determine which numbers are prime:  any composite number will have at least one prime factor less than or equal to its square root.) I've even written out an array of those below 630!

2,3,5      031\       061\       091\       121\       151\       181   

007\011 037\041 067\071 097\101 127\131 157\161 187\191

013\017 043\047 073\077 103\107 133\137 163\167 193\197

019\023 049\053 079\083 109\113 139\143 169\173 199\203

      \029       \059        \089       \119       \149       \179      \209 

211\       241        271\       301\       331\       361\       391\

217\221 247\251 277\281 307\311 337\341 367\371 397\401

223\227 253\257 283\287 313\317 343\347 373\377 403\407

229\233 259\263 289\293 319\323 349\353 379\383 409\413

      \239       \269       \299       \329       \359       \389       \419

421\       451\       481\       511\       541\       571\       601\

427\431 457\461 487\491 517\521 547\551 577\581 607\611

433\437 463\467 493\497 523\527 553\557 583\587 613\617

439\443 469\473 499\503 529\533 559\563 589\593 619\623

      \449       \479       \509       \539       \569       \599       \629

In setting up this array, I've started with the numbers that don't divide 2, 3 or 5, eight numbers to which you keep adding multiples of 30. (The strikethroughs are of composites.) And each row has seven repetitions of these eight numbers, so that the composites of 7 will repeat themselves row by row through adding multiples of 210!  Prime numbers fascinate me...

I've been thinking some more about that problem where you bet money with a 50% chance of losing it and a 50% chance of a 150% profit.  Consider what happens if you bet the same proportion of your money twice and you win once and lose once, as will happen half the time. (Whether you win and then lose or lose and then win doesn't matter:  the result is the same.)  

If the proportion is one sixth,  with a one-quarter gain half the time, you'll end up with a one in twenty-four gain. (This is the proportion that yields the peak profit:  you can prove it through differential calculus.) But if it's 20% (one-fifth), with a 30% gain half the time, you'll  still get a one in twenty-five profit (4%). That's only marginally less, so I'd make that my regular bet:  a bigger bet is likely to get bigger returns overall.  On the other hand, if you raise the proportion to one half, you'll end up with a one-eighth loss, so you shouldn't bet any more than that.  (With a proportion of one third, you'll break even.)

When you're on a winning streak, however, it pays to increase the bet because you can afford greater risk.  I'd start raising it when I've had two wins in a row, leaving me 69% ahead of the level before the first win.  If you raise it to 25%, even if you lose you'll still be 27% ahead, only slightly behind where you were before the second win. But if you win, you'll be just over 130% ahead!

Similarly, after a third win I'd raise it again to 30%.  If you lose, you're 60% ahead; if you win, you're 235% ahead!  And so on until you reach 50%, at which point you should stay until you finally lose. (Of course, that'll only happen when you win eight times in a row...) And after you lose, you go right back to 20% until the next consecutive wins.

The point is that when you have a winning streak, you'll really clean up. (And even your first loss will leave you little worse off than just before your last win.) If you lose six in a row and then win, you'll be down to a third of your earlier level, but if you win six in a row and then lose, you'll have a sixfold increase!

I abhor gambling, but I like to follow in Blaise Pascal's footsteps (that's him in the pic!) and study the odds.