478:
315:
839:
Although Newton correctly calculated the odds of each bet, he provided a separate intuitive explanation to Pepys. He imagined that B and C toss their dice in groups of six, and said that A was most favorable because it required a 6 in only one toss, while B and C required a 6 in each of their tosses.
655:
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160:
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1505:
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1976:
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505:
1130:
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1780:
996:
473:{\displaystyle P(C)=1-\sum _{x=0}^{2}{\binom {18}{x}}\left({\frac {1}{6}}\right)^{x}\left({\frac {5}{6}}\right)^{18-x}={\frac {15166600495229}{25389989167104}}\approx 0.5973\,.}
956:
2268:
1594:
1377:
905:
856:
the probability that each die will select the 6 face when thrown (notice that actually the number of faces of the dice and which face should be selected are irrelevant). If
1554:
1337:
310:{\displaystyle P(B)=1-\sum _{x=0}^{1}{\binom {12}{x}}\left({\frac {1}{6}}\right)^{x}\left({\frac {5}{6}}\right)^{12-x}={\frac {1346704211}{2176782336}}\approx 0.6187\,,}
71:
1599:
1382:
2203:
57:
Pepys initially thought that outcome C had the highest probability, but Newton correctly concluded that outcome A actually has the highest probability.
2833:
1142:
1861:
2171:
2099:
D. Varagnolo, L. Schenato, G. Pillonetto, 2013. "A variation of the Newton–Pepys problem and its connections to size-estimation problems."
2260:
840:
This explanation assumes that a group does not produce more than one 6, so it does not actually correspond to the original problem.
2383:
1279:
As noticed in Rubin and Evans (1961), there are no uniform answers to the generalized Newton–Pepys problem since answers depend on
2808:
2785:
2541:
2489:
2883:
2393:
2131:
650:{\displaystyle P(n)=1-\sum _{x=0}^{n-1}{\binom {6n}{x}}\left({\frac {1}{6}}\right)^{x}\left({\frac {5}{6}}\right)^{6n-x}\,.}
2420:
2519:
2230:
1785:
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2410:
2222:
1068:
1001:
2818:
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2723:
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1733:
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2124:
921:
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2163:
1559:
1342:
863:
2405:
2388:
2373:
2318:
1513:
1296:
677:
38:
corresponded over a problem posed to Pepys by a school teacher named John Smith. The problem was:
2442:
2195:
2813:
2803:
2568:
2531:
2504:
2398:
1287:. There are nonetheless some variations of the previous questions that admit uniform answers:
147:{\displaystyle P(A)=1-\left({\frac {5}{6}}\right)^{6}={\frac {31031}{46656}}\approx 0.6651\,,}
2893:
2860:
2767:
2653:
2648:
2437:
2350:
2325:
2117:
484:
1717:{\displaystyle P(r\geq k;kn_{1},{\frac {1}{n_{1}}})>P(r\geq k;kn_{2},{\frac {1}{n_{2}}})}
1500:{\displaystyle P(r\geq k_{1};k_{1}n,{\frac {1}{n}})>P(r\geq k_{2};k_{2}n,{\frac {1}{n}})}
2774:
2494:
2251:
8:
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2469:
2330:
2002:
2798:
2514:
2427:
2155:
2069:
2051:
2757:
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2588:
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2454:
2340:
1999:
2073:
27:
problem concerning the probability of throwing sixes from a certain number of dice.
2717:
2693:
2669:
2499:
2245:
2061:
2705:
2474:
2368:
2039:
1139:
Notice that, with this notation, the original Newton–Pepys problem reads as: is
2748:
2573:
2556:
2526:
2313:
2065:
52:
C. Eighteen fair dice are tossed independently and at least three "6"s appear.
42:
Which of the following three propositions has the greatest chance of success?
2877:
2793:
2675:
2617:
1269:{\displaystyle P(r\geq 1;6,1/6)\geq P(r\geq 2;12,1/6)\geq P(r\geq 3;18,1/6)}
915:
dice. Then the original Newton–Pepys problem can be generalized as follows:
2681:
2551:
2360:
2335:
2298:
2140:
35:
31:
49:
B. Twelve fair dice are tossed independently and at least two "6"s appear.
2699:
2308:
2187:
1971:{\displaystyle P(r=\nu _{1}k;\nu _{1}n,p)\geq P(r=\nu _{2}k;\nu _{2}n,p)}
24:
487:(although Newton obtained them from first principles). In general, if P(
2823:
2711:
46:
A. Six fair dice are tossed independently and at least one "6" appears.
2056:
2007:
2345:
2286:
2109:
2303:
2291:
2179:
2857:
668:) decreases monotonically towards an asymptotic limit of 1/2.
1997:
2086:
Chaundy, T.W., Bullard, J.E., 1960. "John Smith’s
Problem."
2022:
Chaundy, T.W., Bullard, J.E., 1960. "John Smith’s
Problem."
860:
is the total number of dice selecting the 6 face, then
848:
A natural generalization of the problem is to consider
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1516:
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1345:
1299:
1145:
1071:
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924:
866:
508:
326:
163:
74:
1851:{\displaystyle \nu _{1}\leq \nu _{2},k\leq n,p\in }
1970:
1850:
1774:
1727:(from Varagnolo, Pillonetto and Schenato (2013)):
1716:
1588:
1548:
1499:
1371:
1331:
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1124:
1057:
990:
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676:The solution outlined above can be implemented in
649:
472:
309:
146:
2834:Statal Institute of Higher Education Isaac Newton
578:
560:
385:
372:
222:
209:
2875:
1125:{\displaystyle P(r\geq \nu _{2}k;\nu _{2}n,p)}
1058:{\displaystyle P(r\geq \nu _{1}k;\nu _{1}n,p)}
483:These results may be obtained by applying the
65:The probabilities of outcomes A, B and C are:
2125:
2132:
2118:
491:) is the probability of throwing at least
2055:
2042:(2006). "Isaac Newton as a Probabilist".
2034:
2032:
1993:
1991:
911:correct selections when throwing exactly
643:
466:
303:
140:
2038:
834:
2876:
2029:
1988:
907:is the probability of having at least
2384:Newton's law of universal gravitation
2113:
1998:
1775:{\displaystyle \nu _{1},\nu _{2},n,k}
991:{\displaystyle \nu _{1}\leq \nu _{2}}
2542:Newton's theorem of revolving orbits
2139:
2101:Statistics & Probability Letters
2490:Leibniz–Newton calculus controversy
2231:standing on the shoulders of giants
1290:(from Chaundy and Bullard (1960)):
781:"Probability of at least"
772:# q = Prob( <s sixes in n dice )
13:
1782:are positive natural numbers, and
1556:are positive natural numbers, and
1339:are positive natural numbers, and
843:
564:
376:
213:
14:
2910:
958:be natural positive numbers s.t.
951:{\displaystyle \nu _{1},\nu _{2}}
712:# looking for s = 1, 2 or 3 sixes
2819:Isaac Newton Group of Telescopes
852:non-necessarily fair dice, with
2839:Newton International Fellowship
2520:generalized Gauss–Newton method
2433:Newton's method in optimization
671:
2093:
2080:
2016:
1965:
1921:
1912:
1868:
1845:
1833:
1711:
1663:
1654:
1606:
1589:{\displaystyle n_{1}<n_{2}}
1494:
1446:
1437:
1389:
1372:{\displaystyle k_{1}<k_{2}}
1263:
1231:
1222:
1190:
1181:
1149:
1119:
1075:
1052:
1008:
900:{\displaystyle P(r\geq k;n,p)}
894:
870:
518:
512:
336:
330:
173:
167:
84:
78:
1:
2884:Factorial and binomial topics
1981:
1549:{\displaystyle k,n_{1},n_{2}}
1332:{\displaystyle k_{1},k_{2},n}
730:# ... in n = 6, 12 or 18 dice
2460:Newton's theorem about ovals
7:
2829:Sir Isaac Newton Sixth Form
2485:Corpuscular theory of light
2411:Schrödinger–Newton equation
60:
10:
2915:
2238:Notes on the Jewish Temple
2066:10.1214/088342306000000312
2847:
2784:
2739:
2662:
2604:
2359:
2279:
2214:
2147:
2389:post-Newtonian expansion
2269:Corruptions of Scripture
2261:Ancient Kingdoms Amended
2088:The Mathematical Gazette
2024:The Mathematical Gazette
682:
2579:Absolute space and time
2443:truncated Newton method
2416:Newton's laws of motion
2379:Newton's law of cooling
2814:Isaac Newton Telescope
2804:Isaac Newton Institute
2574:Newton–Puiseux theorem
2569:Parallelogram of force
2557:kissing number problem
2547:Newton–Euler equations
2450:Gauss–Newton algorithm
2399:gravitational constant
2003:"Newton-Pepys Problem"
1972:
1852:
1776:
1718:
1590:
1550:
1501:
1373:
1333:
1270:
1126:
1059:
992:
952:
901:
805:"fair dice:"
651:
556:
474:
368:
311:
205:
148:
55:
2899:Mathematical problems
2768:Isaac Newton Gargoyle
2678: (nephew-in-law)
2654:Copernican Revolution
2649:Scientific Revolution
2510:Newton–Cotes formulas
2374:Newton's inequalities
2351:Structural coloration
1973:
1853:
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1719:
1591:
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1502:
1374:
1334:
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530:
485:binomial distribution
475:
348:
312:
185:
149:
40:
2889:Probability problems
2775:Astronomers Monument
2465:Newton–Pepys problem
2438:Apollonius's problem
2406:Newton–Cartan theory
2319:Newton–Okounkov body
2252:hypotheses non fingo
2241: (c. 1680)
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1600:
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835:Newton's explanation
506:
324:
161:
72:
21:Newton–Pepys problem
2584:Luminiferous aether
2532:Newton's identities
2505:Newton's cannonball
2480:Classical mechanics
2470:Newtonian potential
2331:Newtonian telescope
2044:Statistical Science
16:Probability problem
2809:Isaac Newton Medal
2614: (birthplace)
2428:Newtonian dynamics
2326:Newton's reflector
2103:83 (5), 1472-1478.
2040:Stigler, Stephen M
2000:Weisstein, Eric W.
1968:
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1772:
1714:
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1055:
988:
948:
897:
793:"six in"
647:
470:
307:
144:
2871:
2870:
2763: (sculpture)
2730:Abraham de Moivre
2684: (professor)
2612:Woolsthorpe Manor
2564:Newton's quotient
2537:Newton polynomial
2495:Newton's notation
2226: (1661–1665)
1709:
1652:
1492:
1435:
1065:not smaller than
622:
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295:
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132:
109:
2906:
2859:
2754: (monotype)
2718:William Stukeley
2714: (disciple)
2694:Benjamin Pulleyn
2670:Catherine Barton
2589:Newtonian series
2500:Rotating spheres
2246:General Scholium
2141:Sir Isaac Newton
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2799:Newton's cradle
2780:
2735:
2708: (student)
2706:William Whiston
2702: (student)
2658:
2639:Religious views
2600:
2515:Newton's method
2475:Newtonian fluid
2369:Bucket argument
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919:
865:
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846:
844:Generalizations
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5:
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2779:
2778:
2771:
2764:
2755:
2745:
2743:
2737:
2736:
2734:
2733:
2732: (friend)
2727:
2726: (friend)
2721:
2720: (friend)
2715:
2709:
2703:
2697:
2691:
2690: (mentor)
2688:William Clarke
2685:
2679:
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2644:Occult studies
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2527:Newton fractal
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2507:
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2455:Newton's rings
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2341:Newton's metal
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2314:Newton polygon
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2257:
2248:" (1713;
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2215:Other writings
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2015:
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1583:
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1543:
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1496:
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1471:
1467:
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1434:
1431:
1426:
1423:
1418:
1414:
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1405:
1401:
1397:
1394:
1391:
1388:
1366:
1362:
1358:
1353:
1349:
1328:
1325:
1320:
1316:
1312:
1307:
1303:
1265:
1262:
1258:
1254:
1251:
1248:
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1239:
1236:
1233:
1230:
1227:
1224:
1221:
1217:
1213:
1210:
1207:
1204:
1201:
1198:
1195:
1192:
1189:
1186:
1183:
1180:
1176:
1172:
1169:
1166:
1163:
1160:
1157:
1154:
1151:
1148:
1121:
1118:
1115:
1112:
1107:
1103:
1099:
1096:
1091:
1087:
1083:
1080:
1077:
1074:
1054:
1051:
1048:
1045:
1040:
1036:
1032:
1029:
1024:
1020:
1016:
1013:
1010:
1007:
985:
981:
977:
972:
968:
945:
941:
937:
932:
928:
896:
893:
890:
887:
884:
881:
878:
875:
872:
869:
845:
842:
836:
833:
823:"\n"
683:
673:
670:
658:
657:
646:
640:
637:
634:
631:
626:
621:
618:
613:
606:
601:
596:
593:
588:
580:
575:
571:
568:
562:
554:
551:
548:
543:
540:
537:
533:
529:
526:
523:
520:
517:
514:
511:
481:
480:
469:
465:
462:
457:
456:25389989167104
454:
453:15166600495229
449:
444:
441:
438:
433:
428:
425:
420:
413:
408:
403:
400:
395:
387:
382:
379:
374:
366:
361:
358:
355:
351:
347:
344:
341:
338:
335:
332:
329:
318:
317:
306:
302:
299:
294:
291:
286:
281:
278:
275:
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265:
262:
257:
250:
245:
240:
237:
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224:
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198:
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155:
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143:
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95:
92:
89:
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83:
80:
77:
62:
59:
54:
53:
50:
47:
15:
9:
6:
4:
3:
2:
2911:
2900:
2897:
2895:
2892:
2890:
2887:
2885:
2882:
2881:
2879:
2862:
2858:
2850:
2846:
2840:
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2835:
2832:
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2827:
2825:
2822:
2820:
2817:
2815:
2812:
2810:
2807:
2805:
2802:
2800:
2797:
2795:
2794:Newton (unit)
2792:
2791:
2789:
2787:
2783:
2777:
2776:
2772:
2770:
2769:
2765:
2762:
2760:
2756:
2753:
2751:
2747:
2746:
2744:
2742:
2738:
2731:
2728:
2725:
2724:William Jones
2722:
2719:
2716:
2713:
2710:
2707:
2704:
2701:
2698:
2696: (tutor)
2695:
2692:
2689:
2686:
2683:
2680:
2677:
2676:John Conduitt
2674:
2672: (niece)
2671:
2668:
2667:
2665:
2661:
2655:
2652:
2650:
2647:
2645:
2642:
2640:
2637:
2635:
2632:
2630:
2627:
2625:
2622:
2619:
2618:Cranbury Park
2616:
2613:
2610:
2609:
2607:
2605:Personal life
2603:
2595:
2592:
2591:
2590:
2587:
2585:
2582:
2580:
2577:
2575:
2572:
2570:
2567:
2565:
2562:
2558:
2555:
2554:
2553:
2552:Newton number
2550:
2548:
2545:
2543:
2540:
2538:
2535:
2533:
2530:
2528:
2525:
2521:
2518:
2517:
2516:
2513:
2511:
2508:
2506:
2503:
2501:
2498:
2496:
2493:
2491:
2488:
2486:
2483:
2481:
2478:
2476:
2473:
2471:
2468:
2466:
2463:
2461:
2458:
2456:
2453:
2451:
2448:
2444:
2441:
2439:
2436:
2435:
2434:
2431:
2429:
2426:
2422:
2421:Kepler's laws
2419:
2418:
2417:
2414:
2412:
2409:
2407:
2404:
2400:
2397:
2395:
2394:parameterized
2392:
2390:
2387:
2386:
2385:
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2380:
2377:
2375:
2372:
2370:
2367:
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2282:
2280:Contributions
2278:
2271:
2270:
2266:
2263:
2262:
2258:
2255:
2253:
2247:
2243:
2240:
2239:
2235:
2233:" (1675)
2232:
2228:
2225:
2224:
2220:
2219:
2217:
2213:
2206:
2205:
2201:
2198:
2197:
2193:
2190:
2189:
2185:
2182:
2181:
2177:
2174:
2173:
2169:
2166:
2165:
2161:
2158:
2157:
2153:
2152:
2150:
2146:
2142:
2135:
2130:
2128:
2123:
2121:
2116:
2115:
2112:
2102:
2096:
2089:
2083:
2075:
2071:
2067:
2063:
2058:
2053:
2049:
2045:
2041:
2035:
2033:
2025:
2019:
2010:
2009:
2004:
2001:
1994:
1992:
1987:
1979:
1962:
1959:
1956:
1951:
1947:
1943:
1940:
1935:
1931:
1927:
1924:
1918:
1915:
1909:
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1517:
1508:
1489:
1486:
1481:
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1469:
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1460:
1456:
1452:
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1432:
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1424:
1421:
1416:
1412:
1408:
1403:
1399:
1395:
1392:
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1364:
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1356:
1351:
1347:
1326:
1323:
1318:
1314:
1310:
1305:
1301:
1291:
1288:
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1282:
1277:
1260:
1256:
1252:
1249:
1246:
1243:
1240:
1237:
1234:
1228:
1225:
1219:
1215:
1211:
1208:
1205:
1202:
1199:
1196:
1193:
1187:
1184:
1178:
1174:
1170:
1167:
1164:
1161:
1158:
1155:
1152:
1146:
1137:
1135:
1116:
1113:
1110:
1105:
1101:
1097:
1094:
1089:
1085:
1081:
1078:
1072:
1049:
1046:
1043:
1038:
1034:
1030:
1027:
1022:
1018:
1014:
1011:
1005:
983:
979:
975:
970:
966:
943:
939:
935:
930:
926:
916:
914:
910:
891:
888:
885:
882:
879:
876:
873:
867:
859:
855:
851:
841:
681:
679:
669:
667:
663:
644:
638:
635:
632:
629:
624:
619:
616:
611:
604:
599:
594:
591:
586:
573:
569:
566:
552:
549:
546:
541:
538:
535:
531:
527:
524:
521:
515:
509:
502:
501:
500:
498:
494:
490:
486:
467:
463:
460:
455:
452:
447:
442:
439:
436:
431:
426:
423:
418:
411:
406:
401:
398:
393:
380:
377:
364:
359:
356:
353:
349:
345:
342:
339:
333:
327:
320:
319:
304:
300:
297:
292:
289:
284:
279:
276:
273:
268:
263:
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243:
238:
235:
230:
217:
214:
201:
196:
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190:
186:
182:
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176:
170:
164:
157:
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141:
137:
134:
129:
126:
121:
116:
111:
106:
103:
98:
93:
90:
87:
81:
75:
68:
67:
66:
58:
51:
48:
45:
44:
43:
39:
37:
33:
28:
26:
22:
2894:Isaac Newton
2861:Isaac Newton
2773:
2766:
2758:
2749:
2682:Isaac Barrow
2620: (home)
2464:
2361:Newtonianism
2336:Newton scale
2299:Impact depth
2272: (1754)
2267:
2264: (1728)
2259:
2249:
2236:
2221:
2207: (1711)
2202:
2199: (1707)
2194:
2191: (1704)
2186:
2183: (1704)
2178:
2175: (1687)
2170:
2167: (1684)
2162:
2159: (1671)
2154:
2148:Publications
2100:
2095:
2090:44, 253-260.
2087:
2082:
2057:math/0701089
2047:
2043:
2026:44, 253-260.
2023:
2018:
2006:
1729:
1726:
1509:
1292:
1289:
1284:
1280:
1278:
1138:
1133:
917:
912:
908:
857:
853:
849:
847:
838:
680:as follows:
675:
672:Example in R
665:
661:
659:
499:dice, then:
496:
495:sixes with 6
492:
488:
482:
64:
56:
41:
36:Isaac Newton
32:Samuel Pepys
29:
20:
18:
2761:by Paolozzi
2700:Roger Cotes
2309:Newton disc
2223:Quaestiones
2196:Arithmetica
25:probability
2878:Categories
2848:Categories
2824:XMM-Newton
2741:Depictions
2712:John Keill
2634:Apple tree
2629:Later life
2624:Early life
2204:De Analysi
2050:(3): 400.
1982:References
998:. Is then
293:2176782336
290:1346704211
2663:Relations
2172:Principia
2008:MathWorld
1948:ν
1932:ν
1916:≥
1895:ν
1879:ν
1831:∈
1819:≤
1804:ν
1800:≤
1791:ν
1752:ν
1739:ν
1670:≥
1613:≥
1453:≥
1396:≥
1238:≥
1226:≥
1197:≥
1185:≥
1156:≥
1102:ν
1086:ν
1082:≥
1035:ν
1019:ν
1015:≥
980:ν
976:≤
967:ν
940:ν
927:ν
877:≥
664:grows, P(
636:−
550:−
532:∑
528:−
461:≈
440:−
350:∑
346:−
298:≈
277:−
187:∑
183:−
135:≈
94:−
2786:Namesake
2752:by Blake
2346:Spectrum
2287:Calculus
2256: )
2156:Fluxions
2074:17471221
1132:for all
61:Solution
30:In 1693
2304:Inertia
2292:fluxion
2188:Queries
2180:Opticks
2164:De Motu
1596:, then
1379:, then
1134:n, p, k
2759:Newton
2750:Newton
2072:
739:pbinom
464:0.5973
301:0.6187
138:0.6651
2594:table
2070:S2CID
2052:arXiv
1858:then
130:46656
127:31031
23:is a
1658:>
1574:<
1441:>
1357:<
1283:and
1281:k, n
918:Let
685:for
34:and
19:The
2062:doi
1730:If
1510:If
1293:If
775:cat
660:As
2880::
2068:.
2060:.
2048:21
2046:.
2031:^
2005:.
1990:^
1978:.
1724:.
1507:.
1276:?
1247:18
1206:12
1136:?
748:-1
694:in
437:18
378:18
274:12
215:12
2254:"
2250:"
2244:"
2229:"
2133:e
2126:t
2119:v
2076:.
2064::
2054::
2011:.
1966:)
1963:p
1960:,
1957:n
1952:2
1944:;
1941:k
1936:2
1928:=
1925:r
1922:(
1919:P
1913:)
1910:p
1907:,
1904:n
1899:1
1891:;
1888:k
1883:1
1875:=
1872:r
1869:(
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1843:1
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1837:0
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892:p
889:,
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820:,
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516:n
513:(
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493:n
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468:.
448:=
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427:6
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357:=
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340:=
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334:C
331:(
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285:=
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264:6
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256:(
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239:6
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231:(
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218:x
210:(
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194:=
191:x
180:1
177:=
174:)
171:B
168:(
165:P
142:,
122:=
117:6
112:)
107:6
104:5
99:(
91:1
88:=
85:)
82:A
79:(
76:P
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